Clipping is a technique used to remove portions of lines, polygons, and other primitives that lie outside the visible viewing area or viewport. There are several common clipping algorithms. Cohen-Sutherland line clipping uses bit codes to quickly determine if a line segment can be fully accepted or rejected for clipping. Sutherland-Hodgman polygon clipping considers each viewport edge individually, clips the polygon against that edge plane, and generates a new clipped polygon. Perspective projection transforms 3D objects to 2D screen coordinates, and clipping must account for objects behind the viewer; this can be done by clipping in camera coordinates before perspective projection or in homogeneous screen coordinates after projection.
Polygon clipping involves taking a polygon and clipping it against another shape to produce one or more smaller polygons. The Sutherland-Hodgman algorithm handles polygon clipping by testing each edge of the clipping polygon against each edge of the clip shape. There are four cases for how an edge can be clipped - wholly inside, exit, wholly outside, enter - and the algorithm saves or discards vertices based on these cases. Repeatedly clipping against each edge of the clip shape handles all cases and produces the final clipped polygon(s).
This document discusses different techniques for computer graphics clipping. It describes point clipping, line clipping using the Cohen-Sutherland and Liang-Barsky algorithms, area/polygon clipping using the Sutherland-Hodgman and Weiler-Atherton algorithms, curve clipping, and text clipping. Various preliminary tests and intersection calculations are used to identify and remove graphic elements that are outside the clipping region.
The document discusses curve and text clipping techniques. It describes three methods for curve clipping: fully in, fully out, and partially in the clip window. It also outlines three approaches to text clipping: all-or-nothing string clipping, all-or-nothing character clipping, and individual character clipping where parts of characters outside the window are clipped off. The document was written by Arvind Kumar on curve and text clipping algorithms.
The document discusses the Sutherland-Hodgeman polygon clipping algorithm. It clips a polygon by considering it against each boundary edge of the window. It passes the polygon's vertices to clipping procedures for the left, right, bottom and top edges. At each stage, it generates a new set of vertices for the clipped polygon which is passed to the next stage. There are four cases considered - when vertices are inside/outside the window boundary and their intersection points are determined and stored in the output vertex list. Once all vertices are clipped against one boundary, the result is passed to the next boundary for further clipping until the fully clipped polygon is produced.
The document discusses B-spline curves, which are defined using control points, knots, and basis functions. B-spline curves allow for local control of the curve shape and independent specification of the degree and number of control points. The curve is computed as a weighted sum of basis functions evaluated over the knot vector. The shape of the B-spline curve can be modified by changing the control points, knots, or degree of the curve.
Polygon clipping with sutherland hodgeman algorithm and scan line fill algorithmMani Kanth
This document discusses two computer graphics algorithms: the Sutherland-Hodgman polygon clipping algorithm and the scan line fill algorithm. The Sutherland-Hodgman algorithm clips polygons against the boundaries of a clip window by testing each polygon edge against each clip window edge. The scan line fill algorithm fills polygons by drawing between intersection points of polygon edges and horizontal scan lines from top to bottom. It handles polygon corners and edge endpoints to correctly determine the interior and exterior regions.
The Weiler-Atherton algorithm clips concave polygons by tracing intersection points between the subject and clip polygons. It processes vertices clockwise, following the subject polygon boundary for outside-inside vertex pairs and the clip polygon boundary for inside-outside pairs. The algorithm traces the polygons, records intersection points, and clips the subject polygon accordingly until it reaches the starting vertex.
Polygon clipping involves taking a polygon and clipping it against another shape to produce one or more smaller polygons. The Sutherland-Hodgman algorithm handles polygon clipping by testing each edge of the clipping polygon against each edge of the clip shape. There are four cases for how an edge can be clipped - wholly inside, exit, wholly outside, enter - and the algorithm saves or discards vertices based on these cases. Repeatedly clipping against each edge of the clip shape handles all cases and produces the final clipped polygon(s).
This document discusses different techniques for computer graphics clipping. It describes point clipping, line clipping using the Cohen-Sutherland and Liang-Barsky algorithms, area/polygon clipping using the Sutherland-Hodgman and Weiler-Atherton algorithms, curve clipping, and text clipping. Various preliminary tests and intersection calculations are used to identify and remove graphic elements that are outside the clipping region.
The document discusses curve and text clipping techniques. It describes three methods for curve clipping: fully in, fully out, and partially in the clip window. It also outlines three approaches to text clipping: all-or-nothing string clipping, all-or-nothing character clipping, and individual character clipping where parts of characters outside the window are clipped off. The document was written by Arvind Kumar on curve and text clipping algorithms.
The document discusses the Sutherland-Hodgeman polygon clipping algorithm. It clips a polygon by considering it against each boundary edge of the window. It passes the polygon's vertices to clipping procedures for the left, right, bottom and top edges. At each stage, it generates a new set of vertices for the clipped polygon which is passed to the next stage. There are four cases considered - when vertices are inside/outside the window boundary and their intersection points are determined and stored in the output vertex list. Once all vertices are clipped against one boundary, the result is passed to the next boundary for further clipping until the fully clipped polygon is produced.
The document discusses B-spline curves, which are defined using control points, knots, and basis functions. B-spline curves allow for local control of the curve shape and independent specification of the degree and number of control points. The curve is computed as a weighted sum of basis functions evaluated over the knot vector. The shape of the B-spline curve can be modified by changing the control points, knots, or degree of the curve.
Polygon clipping with sutherland hodgeman algorithm and scan line fill algorithmMani Kanth
This document discusses two computer graphics algorithms: the Sutherland-Hodgman polygon clipping algorithm and the scan line fill algorithm. The Sutherland-Hodgman algorithm clips polygons against the boundaries of a clip window by testing each polygon edge against each clip window edge. The scan line fill algorithm fills polygons by drawing between intersection points of polygon edges and horizontal scan lines from top to bottom. It handles polygon corners and edge endpoints to correctly determine the interior and exterior regions.
The Weiler-Atherton algorithm clips concave polygons by tracing intersection points between the subject and clip polygons. It processes vertices clockwise, following the subject polygon boundary for outside-inside vertex pairs and the clip polygon boundary for inside-outside pairs. The algorithm traces the polygons, records intersection points, and clips the subject polygon accordingly until it reaches the starting vertex.
The Sutherland-Hodgman algorithm clips polygons by clipping against each edge of the clipping window in a specific order: left, top, right, bottom. It works by testing each edge of the polygon against the clipping window boundary and either keeping or discarding vertices based on whether they are inside or outside the window. The algorithm results in a clipped polygon that only includes vertices and edge intersections that are inside the clipping window.
The sutherland hodgeman polygon clipping algorithmMani Kanth
The document discusses the Sutherland Hodgeman polygon clipping algorithm. It is used to clip a polygon by specifying a clipping window. The algorithm clips the vertices of the polygon against each edge of the clipping window by finding intersection points. If an edge is not completely inside the clipping window, the portion outside is clipped off. An example is provided to demonstrate clipping a polygon ABCDE against a clipping window PQRS.
Sutherland hodgman polygon clipping algorithmTawfiq Ahmed
Sutherland Hodgman polygon clipping algorithm is a very simple clipping algorithm to understand. I hope my slide will help you guys.
-Thanks
Tawfiq Ahmed
The document describes the Breshenham's circle generation algorithm. It explains that the algorithm uses a decision parameter to iteratively select pixels along the circumference of a circle. It provides pseudocode for the algorithm, which initializes x and y values, calculates a decision parameter, and increments x while decrementing y at each step, plotting points based on the decision parameter. An example of applying the algorithm to generate a circle with radius 5 is also provided.
Raster Scan Graphics, Line Drawing Algorithm and Circle Drawing Algorithm Aparna Joshi
This presentation contains -
- Characteristics of line drawing algorithm
- DDA Line Drawing Algorithm
- Bresenham's Line Drawing Algorithm
- Circle Drawing Algorithm
- Eight way symmetry of a circle
- Breseham's Circle Drawing Algorithm
- Midpoint Circle Algorithm
The document discusses the Liang-Barsky line clipping algorithm, an algorithm used to clip lines to a rectangular viewing area. It is covered by Arvind Kumar, an assistant professor at Vidya College of Engineering. As an example, the line clipping algorithm is shown clipping a line with endpoints (22.5,15) and (25,16).
A Bezier curve is a mathematically defined curve used in graphic applications. It is defined by four points: two anchors at the initial and terminal positions and two handles that control the shape of the curve. The curve can be altered by moving the handles. Bezier curves are commonly used to piece together complex curves from simpler component curves. Matching endpoints ensures continuity, while aligning handles ensures smooth tangency between connecting segments.
Curve clipping involves using polygon clipping to test if a curved object's bounding rectangle overlaps a clipping window. If there is no overlap, the object is discarded. If there is overlap, the simultaneous curve and boundary equations are solved to find intersection points. Special cases like circles are considered, such as discarding a circle if its center is outside the clipping window plus/minus the radius. Bezier and spline curves can also be clipped by approximating them as polylines or using their convex hull properties.
The Cyrus-Beck algorithm is used for line clipping against non-rectangular convex polygons. It uses a parametric equation to find the intersection point of the line with the polygon boundary. The algorithm calculates the time values for the line endpoints at each polygon edge, then uses those times in the parametric equation to find the clipped line segment P'0 and P'1 that is visible within the polygon clipping window.
a spline is a flexible strip used to produce a smooth curve through a designated set of points.
Polynomial sections are fitted so that the curve passes through each control point, Resulting curve is said to interpolate the set of control points.
Clipping is a process that extracts portions of data or scenes inside a specified clipping region. It uses endpoint codes, which assign a 4-bit code to line endpoints to indicate if they are inside or outside the clipping window. One algorithm is the Cohen-Sutherland algorithm which uses these endpoint codes to test if lines are completely inside, completely outside, or intersect the clipping window. Another is the Mid-Point Subdivision algorithm which avoids directly calculating line-window intersections by performing a binary search via dividing lines at their midpoint.
Clipping is a technique that identifies parts of an image that are inside or outside a defined clipping region or window. There are different types of clipping including point, line, polygon, curve, and text clipping. The Cohen-Sutherland algorithm is commonly used for line clipping. It assigns 4-bit codes to line endpoints to determine if they are fully inside, outside, or intersect the clipping window boundary. Intersecting line segments are then subdivided and clipped. Midpoint subdivision is another algorithm that divides partially visible lines at their midpoint into shorter segments.
The document discusses the Cohen-Sutherland line clipping algorithm. It describes how the algorithm performs initial tests on a line to determine if intersection calculations are needed by checking for trivial acceptance or rejection of line segments. If a line cannot be fully accepted or rejected, it is divided at a clip edge and the process is repeated iteratively until the line is fully processed. An example of applying the algorithm to clip a line against a rectangle is also provided.
The document discusses the convex hull algorithm. It begins by defining a convex hull as the shape a rubber band would take if stretched around pins on a board. It then provides explanations of extreme points, edges, and applications of convex hulls. Various algorithms for finding convex hulls are presented, including divide and conquer in O(n log n) time and Jarvis march in O(n^2) time in the worst case.
The document discusses the 3D viewing pipeline which transforms 3D world coordinates to 2D viewport coordinates through a series of steps. It then describes parallel and perspective projections. Parallel projection preserves object scale and shape while perspective projection does not due to foreshortening effects. Perspective projection involves projecting 3D points along projection rays to a view plane based on a center of projection. Other topics covered include vanishing points, different types of perspective projections, and how viewing parameters affect the view volume and object positioning in the view plane coordinates.
Jack Bresenham developed an efficient algorithm for drawing lines on a raster display. The Bresenham's line algorithm uses only integer arithmetic to determine the next pixel to plot, allowing fast computation. It works by calculating a decision parameter to choose either the upper or lower pixel as it moves from the starting to ending point of the line. The algorithm guarantees connected lines and plots each point exactly once for accurate rendering compared to other methods.
There are two main types of projections: perspective and parallel. In perspective projection, lines converge to a single point called the center of projection, creating the illusion of depth. In parallel projection, lines remain parallel as they are projected onto the view plane. Perspective projection is more realistic but parallel projection preserves proportions. Perspective projections can be one-point, two-point, or three-point depending on the number of principal vanishing points. Parallel projections include orthographic and oblique views. Common orthographic views are elevations and plans. Axonometric views show multiple sides.
Cohen-Sutherland Line Clipping Algorithm:
When drawing a 2D line on screen, it might happen that one or both of the endpoints are outside the screen while a part of the line should still be visible. In that case, an efficient algorithm is needed to find two new endpoints that are on the edges on the screen, so that the part of the line that's visible can now be drawn. This way, all those points of the line outside the screen are clipped away and you don't need to waste any execution time on them.
A good clipping algorithm is the Cohen-Sutherland algorithm for this solution.
By,
Maruf Abdullah Rion
The document discusses various graphics concepts used in C programming including:
1. Color constants, fill patterns, graphics drivers, errors, and modes that are used in graphics functions.
2. The functions closegraph, detectgraph, getbkcolor, getcolor, getmaxx, getmaxy, getpixel, getx, gety, grapherrormsg, graphresult, initgraph, outtext, outtexttxy, putpixel, setbkcolor, setcolor, and setfillpattern and their purposes.
3. initgraph initializes the graphics system by loading a driver and putting the system into graphics mode, while closegraph deallocates memory and restores the original screen mode.
The Sutherland-Hodgman algorithm clips polygons by clipping against each edge of the clipping window in a specific order: left, top, right, bottom. It works by testing each edge of the polygon against the clipping window boundary and either keeping or discarding vertices based on whether they are inside or outside the window. The algorithm results in a clipped polygon that only includes vertices and edge intersections that are inside the clipping window.
The sutherland hodgeman polygon clipping algorithmMani Kanth
The document discusses the Sutherland Hodgeman polygon clipping algorithm. It is used to clip a polygon by specifying a clipping window. The algorithm clips the vertices of the polygon against each edge of the clipping window by finding intersection points. If an edge is not completely inside the clipping window, the portion outside is clipped off. An example is provided to demonstrate clipping a polygon ABCDE against a clipping window PQRS.
Sutherland hodgman polygon clipping algorithmTawfiq Ahmed
Sutherland Hodgman polygon clipping algorithm is a very simple clipping algorithm to understand. I hope my slide will help you guys.
-Thanks
Tawfiq Ahmed
The document describes the Breshenham's circle generation algorithm. It explains that the algorithm uses a decision parameter to iteratively select pixels along the circumference of a circle. It provides pseudocode for the algorithm, which initializes x and y values, calculates a decision parameter, and increments x while decrementing y at each step, plotting points based on the decision parameter. An example of applying the algorithm to generate a circle with radius 5 is also provided.
Raster Scan Graphics, Line Drawing Algorithm and Circle Drawing Algorithm Aparna Joshi
This presentation contains -
- Characteristics of line drawing algorithm
- DDA Line Drawing Algorithm
- Bresenham's Line Drawing Algorithm
- Circle Drawing Algorithm
- Eight way symmetry of a circle
- Breseham's Circle Drawing Algorithm
- Midpoint Circle Algorithm
The document discusses the Liang-Barsky line clipping algorithm, an algorithm used to clip lines to a rectangular viewing area. It is covered by Arvind Kumar, an assistant professor at Vidya College of Engineering. As an example, the line clipping algorithm is shown clipping a line with endpoints (22.5,15) and (25,16).
A Bezier curve is a mathematically defined curve used in graphic applications. It is defined by four points: two anchors at the initial and terminal positions and two handles that control the shape of the curve. The curve can be altered by moving the handles. Bezier curves are commonly used to piece together complex curves from simpler component curves. Matching endpoints ensures continuity, while aligning handles ensures smooth tangency between connecting segments.
Curve clipping involves using polygon clipping to test if a curved object's bounding rectangle overlaps a clipping window. If there is no overlap, the object is discarded. If there is overlap, the simultaneous curve and boundary equations are solved to find intersection points. Special cases like circles are considered, such as discarding a circle if its center is outside the clipping window plus/minus the radius. Bezier and spline curves can also be clipped by approximating them as polylines or using their convex hull properties.
The Cyrus-Beck algorithm is used for line clipping against non-rectangular convex polygons. It uses a parametric equation to find the intersection point of the line with the polygon boundary. The algorithm calculates the time values for the line endpoints at each polygon edge, then uses those times in the parametric equation to find the clipped line segment P'0 and P'1 that is visible within the polygon clipping window.
a spline is a flexible strip used to produce a smooth curve through a designated set of points.
Polynomial sections are fitted so that the curve passes through each control point, Resulting curve is said to interpolate the set of control points.
Clipping is a process that extracts portions of data or scenes inside a specified clipping region. It uses endpoint codes, which assign a 4-bit code to line endpoints to indicate if they are inside or outside the clipping window. One algorithm is the Cohen-Sutherland algorithm which uses these endpoint codes to test if lines are completely inside, completely outside, or intersect the clipping window. Another is the Mid-Point Subdivision algorithm which avoids directly calculating line-window intersections by performing a binary search via dividing lines at their midpoint.
Clipping is a technique that identifies parts of an image that are inside or outside a defined clipping region or window. There are different types of clipping including point, line, polygon, curve, and text clipping. The Cohen-Sutherland algorithm is commonly used for line clipping. It assigns 4-bit codes to line endpoints to determine if they are fully inside, outside, or intersect the clipping window boundary. Intersecting line segments are then subdivided and clipped. Midpoint subdivision is another algorithm that divides partially visible lines at their midpoint into shorter segments.
The document discusses the Cohen-Sutherland line clipping algorithm. It describes how the algorithm performs initial tests on a line to determine if intersection calculations are needed by checking for trivial acceptance or rejection of line segments. If a line cannot be fully accepted or rejected, it is divided at a clip edge and the process is repeated iteratively until the line is fully processed. An example of applying the algorithm to clip a line against a rectangle is also provided.
The document discusses the convex hull algorithm. It begins by defining a convex hull as the shape a rubber band would take if stretched around pins on a board. It then provides explanations of extreme points, edges, and applications of convex hulls. Various algorithms for finding convex hulls are presented, including divide and conquer in O(n log n) time and Jarvis march in O(n^2) time in the worst case.
The document discusses the 3D viewing pipeline which transforms 3D world coordinates to 2D viewport coordinates through a series of steps. It then describes parallel and perspective projections. Parallel projection preserves object scale and shape while perspective projection does not due to foreshortening effects. Perspective projection involves projecting 3D points along projection rays to a view plane based on a center of projection. Other topics covered include vanishing points, different types of perspective projections, and how viewing parameters affect the view volume and object positioning in the view plane coordinates.
Jack Bresenham developed an efficient algorithm for drawing lines on a raster display. The Bresenham's line algorithm uses only integer arithmetic to determine the next pixel to plot, allowing fast computation. It works by calculating a decision parameter to choose either the upper or lower pixel as it moves from the starting to ending point of the line. The algorithm guarantees connected lines and plots each point exactly once for accurate rendering compared to other methods.
There are two main types of projections: perspective and parallel. In perspective projection, lines converge to a single point called the center of projection, creating the illusion of depth. In parallel projection, lines remain parallel as they are projected onto the view plane. Perspective projection is more realistic but parallel projection preserves proportions. Perspective projections can be one-point, two-point, or three-point depending on the number of principal vanishing points. Parallel projections include orthographic and oblique views. Common orthographic views are elevations and plans. Axonometric views show multiple sides.
Cohen-Sutherland Line Clipping Algorithm:
When drawing a 2D line on screen, it might happen that one or both of the endpoints are outside the screen while a part of the line should still be visible. In that case, an efficient algorithm is needed to find two new endpoints that are on the edges on the screen, so that the part of the line that's visible can now be drawn. This way, all those points of the line outside the screen are clipped away and you don't need to waste any execution time on them.
A good clipping algorithm is the Cohen-Sutherland algorithm for this solution.
By,
Maruf Abdullah Rion
The document discusses various graphics concepts used in C programming including:
1. Color constants, fill patterns, graphics drivers, errors, and modes that are used in graphics functions.
2. The functions closegraph, detectgraph, getbkcolor, getcolor, getmaxx, getmaxy, getpixel, getx, gety, grapherrormsg, graphresult, initgraph, outtext, outtexttxy, putpixel, setbkcolor, setcolor, and setfillpattern and their purposes.
3. initgraph initializes the graphics system by loading a driver and putting the system into graphics mode, while closegraph deallocates memory and restores the original screen mode.
The document discusses different concepts related to clipping in computer graphics including 2D and 3D clipping. It describes how clipping is used to eliminate portions of objects that fall outside the viewing frustum or clip window. Various clipping techniques are covered such as point clipping, line clipping, polygon clipping, and the Cohen-Sutherland algorithm for 2D region clipping. The key purposes of clipping are to avoid drawing objects that are not visible, improve efficiency by culling invisible geometry, and prevent degenerate cases.
CAD - Unit-1 (Fundamentals of Computer Graphics)Priscilla CPG
This document provides an overview of computer-aided design (CAD). It discusses the different types of CAD (2D, 2.5D, and 3D) and how CAD software is used to create and test models. CAD is used in fields like architecture, engineering, and medical design. The document then covers the product design cycle and how CAD/CAM fits within stages like synthesis, analysis, and manufacturing. It also discusses concurrent engineering and the benefits of a collaborative design process. Finally, it explains fundamental CAD concepts like transformations, viewing, clipping algorithms, and the Sutherland-Hodgman area clipping method.
Computer graphics involves the creation, manipulation and storage of geometric objects and images. It has various applications including computer-aided design, presentation graphics, computer art, entertainment, education and training, scientific visualization, image processing, and graphical user interfaces. Graphics packages provide programmatic access to graphics functions and libraries for tasks like 2D drawing, modeling, and rendering.
1. Clipping is a procedure that identifies parts of an image that are inside or outside a specified region, called the clip window. Parts inside the window are displayed, while outside parts are discarded.
2. There are different types of clipping like point, curve, text, and line clipping. Line clipping involves testing if line segments are fully inside/outside the window, and calculating intersections if they cross window boundaries.
3. Popular line clipping algorithms like Cohen-Sutherland and Liang-Barsky assign codes to line endpoints to quickly determine if lines are fully in/out of the window without calculating intersections. They find intersection points to clip lines that cross window edges.
Raster scan systems use a video controller to refresh the screen by accessing pixels stored in a frame buffer in memory. The video controller uses two registers to iterate through each pixel location, retrieving the pixel value and using it to set the intensity of the CRT beam. It draws one scan line at a time from top to bottom until the entire screen is refreshed at a rate of 60 frames per second. Display processors can offload graphics processing tasks from the CPU by performing operations like scan conversion and generating lines and color areas to draw objects in the frame buffer.
Cohen-sutherland & liang-basky line clipping algorithmShilpa Hait
The document describes two line clipping algorithms: Cohen-Sutherland and Liang-Barsky. Cohen-Sutherland assigns region codes to line endpoints and checks for complete visibility, invisibility, or partial visibility. It then finds intersection points if the line is partially visible. Liang-Barsky uses the parametric line equation and clipping window inequalities to determine intersection points u1 and u2, clipping the line between those points if u1 < u2. Liang-Barsky is generally more efficient as it requires only one division to update u1 and u2, while Cohen-Sutherland may repeatedly calculate unnecessary intersections.
The document discusses different algorithms for polygon clipping, which is a process that identifies the visible portions of a polygon through a clipping window. It describes the Sutherland-Hodgeman algorithm, which clips polygons by extending the edges of a convex clip polygon and selecting only visible vertices. The Weiler-Atherton algorithm modifies this approach to correctly display concave polygons. Polygon clipping is important for video games to maximize frame rate by avoiding rendering calculations for invisible portions of polygons.
A polygon is a closed two-dimensional shape with straight or curved sides. It can be defined by an ordered sequence of vertices and edges connecting consecutive vertices. The scan line polygon fill algorithm uses an odd-even rule to determine if a point is inside or outside the polygon by counting edge crossings along a scan line from that point to infinity. Boundary fill and flood fill are two area filling algorithms that color the interior of a polygon or region by recursively filling neighboring pixels of the same color.
This document discusses various algorithms for polygon scan conversion and filling, including:
- The scan line polygon fill algorithm which determines pixel color by calculating polygon edge intersections with scan lines and using an odd-even rule.
- Methods for handling special cases like horizontal edges and vertex intersections.
- Using a sorted edge table and active edge list to incrementally calculate edge intersections across scan lines.
- Flood fill and depth/z-buffer algorithms for hidden surface removal when rendering overlapping polygons.
Clipping algorithms identify portions of an image that are inside or outside a specified clipping region. They are used to extract a defined scene for viewing, identify visible surfaces, and perform other drawing and display operations. Common types of clipping include point, line, polygon, and curve clipping. Algorithms like Cohen-Sutherland and mid-point subdivision use codes and binary subdivision to efficiently determine which image portions are visible and should be displayed.
This document discusses different techniques for clipping graphics objects to a viewing window, including:
- Point and line clipping, where points and line segments outside the window are "clipped" and not drawn.
- The Cohen-Sutherland and Liang-Barsky line clipping algorithms, which use outcodes and parametric equations to efficiently determine which portions of lines are inside the window.
- Area clipping techniques like the Sutherland-Hodgman algorithm, which successively clips a polygon against each boundary of the window by comparing vertices and finding intersection points.
This lecture discusses different clipping algorithms. Clipping removes portions of primitives like points, lines, and polygons that lie outside a region of interest. It describes algorithms for point clipping, line clipping including the Cohen-Sutherland algorithm, and polygon clipping using the Sutherland-Hodgman algorithm. The Sutherland-Hodgman algorithm clips polygons by processing each polygon edge against window edges and handling four different cases that may result.
The document discusses different techniques for thickening lines and clipping primitives during rasterization. It describes Cohen-Sutherland and Sutherland-Hodgman algorithms for clipping lines and polygons against viewing windows. Clipping removes parts of primitives that fall outside the viewport, reducing unnecessary computation during rasterization. Various approaches are discussed for thickening lines during drawing and clipping lines, polygons, and other primitives efficiently.
The document discusses different techniques for thickening lines and clipping primitives during rasterization. It describes Cohen-Sutherland and Sutherland-Hodgman algorithms for clipping lines and polygons against viewing windows. Clipping removes parts of primitives that fall outside the viewport, reducing unnecessary computation during rasterization. Various approaches are discussed for thickening lines during drawing and clipping lines, polygons, and other primitives efficiently.
Clipping is the process of removing graphics that are outside the viewing area. There are different types of clipping including point, line, area/polygon, curve, and text clipping. Line clipping algorithms like Cohen-Sutherland and Liang-Barsky use line endpoints and intersection calculations to determine which portions of a line are inside the clip region. Polygon clipping algorithms like Sutherland-Hodgman and Weiler-Atherton process polygon vertices and edges to generate clipped polygon areas. Curve and text clipping also use bounding areas and intersection calculations to clip complex graphical elements.
Clipping polygons is more complex than clipping lines because it involves considering the polygon as a whole rather than just its individual line segments. The Sutherland-Hodgman algorithm clips polygons by considering each edge of the clipping region individually and clipping the subject polygon against that edge, resulting in a new clipped polygon. This process is repeated for each clipping region edge until the polygon is fully clipped. The algorithm takes as input a list of polygon vertices and outputs a new list of clipped polygon vertices.
The Lian-Barsky algorithm is a line clipping algorithm. This algorithm is more efficient than Cohen–Sutherland line clipping algorithm and can be extended to 3-Dimensional clipping. This algorithm is considered to be the faster parametric line-clipping algorithm. The following concepts are used in this clipping:
The parametric equation of the line.
The inequalities describing the range of the clipping window which is used to determine the intersections between the line and the clip window.
The document discusses various 2D geometric transformations including reflections, shears, and their matrix representations. It also covers 2D viewing pipelines including coordinate systems, clipping techniques like Cohen-Sutherland and Liang-Barsky line clipping, and Sutherland-Hodgeman polygon clipping. Reflections are described as 180 degree rotations about an axis. Shearing shifts points along an axis proportional to their coordinate on the other axis. Clipping algorithms discard or shorten line/polygon segments that fall outside the clip region.
he capability that show some part of object internal a specify window is called windowing and a rectangular region in a world coordinate system is called window. ... Points and lines which are outside the window are "cut off" from view. This process of "cutting off" parts of the image of the world is called Clipping.
The document discusses different techniques for clipping lines and polygons to a viewing window or clipping region.
It describes line clipping algorithms like Cohen-Sutherland that use outcodes to quickly reject lines outside the clipping region or clip lines intersecting the boundary. It also discusses the midpoint subdivision algorithm for line clipping.
For polygon clipping, it explains the Sutherland-Hodgeman algorithm which clips polygons against each window edge one by one, dividing the polygon into smaller clipped polygons inside the viewing region.
The document discusses different types of clipping techniques used in computer graphics. It describes clipping as identifying portions of an image that are inside or outside a specified region. There are different types of clipping including point, line, area/polygon, curve, and text clipping. Line clipping algorithms like Cohen-Sutherland and Liang-Barsky are described. Polygon clipping uses the Sutherland-Hodgeman algorithm. Window-to-viewport coordinate transformation maps a window region to a viewport using scaling and translation to maintain relative proportions.
The document discusses algorithms for finding the convex hull of a set of points in two dimensions. It describes the Jarvis march (also called the gift wrapping algorithm) and the Graham scan algorithm. The Jarvis march finds the convex hull by iterating through points and finding the next point that forms the smallest interior angle with the last two points on the hull. The Graham scan sorts points by angle and then iterates through, eliminating any point that forms an obtuse angle with the last three points. Both algorithms run in O(n log n) time.
The document discusses techniques for clipping and rasterization in computer graphics. It covers line segment clipping algorithms like Cohen-Sutherland and Liang-Barsky. It also discusses polygon clipping, including brute force, triangulation, and a black box pipeline approach. Finally, it covers rasterization techniques for points, lines, and polygons, including inside-outside testing methods, fill algorithms like flood fill and scanline fill.
The document discusses different algorithms for clipping polygons and lines to a viewing window, including the Sutherland-Hodgman and Weiler-Atherton polygon clipping algorithms. The Sutherland-Hodgman algorithm clips polygons by processing edges against each window boundary edge but can result in disconnected line segments or extraneous lines for concave polygons. The Weiler-Atherton algorithm addresses this by following either the polygon or window boundary depending on if the vertex pair is outside to inside or vice versa.
The document discusses different algorithms for line, polygon, and text clipping:
1. The Cyrus-Beck algorithm clips lines by finding their intersections with clip boundaries and using the sign of the denominator to determine if the point is entering or leaving the clip region.
2. Polygon clipping algorithms like Sutherland-Hodgeman and Weiler-Atherton successively clip polygons against boundary edges, outputting new edge points. Weiler-Atherton correctly handles concave polygons.
3. Text and curve clipping can also be performed using similar techniques to the ones used for lines and polygons.
The document discusses algorithms for polygon rasterization and clipping. The scanline algorithm is described for filling polygons by processing one scanline at a time from top to bottom. Triangle rasterization simplifies this by only processing triangles. Line clipping and the Cohen-Sutherland algorithm are explained for clipping lines to a rectangular viewport. The Sutherland-Hodgeman algorithm generalizes this to polygon clipping by following polygon edges and switching to viewport edges when the polygon exits the viewport. 3D clipping extends these ideas by treating near and far clipping planes similarly to 2D plane clipping.
Clipping is a process used in computer graphics to determine the visible and invisible portions of an image or object when only part of it can be displayed in the viewing area. It works by extracting the desired visible portion and discarding anything outside the viewing area. Some key applications of clipping include identifying visible areas of 3D objects, creating objects using solid modeling, and drawing and manipulation operations. The Cohen-Sutherland algorithm is commonly used for line clipping, where it assigns region codes to line endpoints and finds the intersection points of any partially visible lines to clip them to the visible region.
The document discusses several algorithms for computing the 3D convex hull of a set of points: gift wrapping, divide and conquer, and incremental. Gift wrapping directly extends the 2D algorithm and has complexity O(nF) where F is the number of faces. Divide and conquer recursively splits the point set into halves, computes the convex hulls, and merges them in O(n log n) time. The incremental algorithm maintains a conflict graph to efficiently find the visible faces from each new point in O(n log n) time.
This document discusses different types of clipping techniques used in computer graphics. It focuses on line clipping, describing the Cohen-Sutherland line clipping algorithm. This algorithm assigns region codes to line endpoints to determine if the line is fully inside, outside or intersecting the clipping window. It then calculates intersections with window boundaries to clip the line segments inside the window.
Sutherlands Cohen and Hodgeman algorithmsRohit Jain
The document discusses different algorithms for clipping graphics objects to a viewing window:
- The Cohen-Sutherland algorithm clips lines efficiently by assigning region codes to endpoints and clipping any lines where the endpoints have a common set bit in their codes.
- The Sutherland-Hodgman algorithm clips areas by comparing polygons to each boundary in turn, saving vertices that fall inside each boundary for the next clip.
- Clipping graphics objects ensures only those within the window are drawn, improving efficiency over drawing all objects in a scene.
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3. Recap: Homogeneous Coords
3
Intuitively:
The w coordinate of a homogeneous point is
typically 1
Decreasing w makes the point “bigger”, meaning further from
the origin
Homogeneous points with w = 0 are thus “points at infinity”,
meaning infinitely far away in some direction. (What
direction?)
To help illustrate this, imagine subtracting two homogeneous
points: the result is (as expected) a vector
5. Recap: Perspective Projection
5
The geometry of the situation:
Desired
result:
P (x, y, z)X
Z
View
plane
d
(0,0,0) x’ = ?
' , ' ,
d x x d y y
x y z d
z z d z z d
× ×
= = = = =
6. Recap: Perspective Projection Matrix
6
Example:
Or, in 3-D coordinates:
=
10100
0100
0010
0001
z
y
x
ddz
z
y
x
d
dz
y
dz
x
,,
7. Recap: OpenGL’s Persp. Proj. Matrix
7
OpenGL’s gluPerspective() command generates
a slightly more complicated matrix:
Can you figure out what this matrix does?
=
−
−
××
−
+
2
cotwhere
0100
2
00
000
000
y
farnear
nearfar
farnear
nearfar
fov
f
ZZ
ZZ
ZZ
ZΖ
f
aspect
f
8. Projection Matrices
8
Now that we can express perspective foreshortening
as a matrix, we can composite it onto our other
matrices with the usual matrix multiplication
End result: can create a single matrix encapsulating
modeling, viewing, and projection transforms
Though you will recall that in practice OpenGL separates the
modelview from projection matrix (why?)
10. Next Topic: Clipping
10
We’ve been assuming that all primitives (lines,
triangles, polygons) lie entirely within the viewport
In general, this assumption will not hold
12. Why Clip?
12
Bad idea to rasterize outside of framebuffer bounds
Also, don’t waste time scan converting pixels outside
window
13. Clipping
13
The naïve approach to clipping lines:
for each line segment
for each edge of viewport
find intersection points
pick “nearest” point
if anything is left, draw it
What do we mean by “nearest”?
How can we optimize this?
14. Trivial Accepts
14
Big optimization: trivial accept/rejects
How can we quickly determine whether a line
segment is entirely inside the viewport?
A: test both endpoints.
xmin xmax
ymax
ymin
15. Trivial Rejects
15
How can we know a line is outside viewport?
A: if both endpoints on wrong side of same edge, can
trivially reject line
xmin xmax
ymax
ymin
17. Cohen-Sutherland Line Clipping
17
Divide viewplane into regions defined by viewport
edges
Assign each region a 4-bit outcode:
0000 00100001
1001
0101 0100
1000 1010
0110
xmin xmax
ymax
ymin
18. Cohen-Sutherland Line Clipping
18
To what do we assign outcodes?
How do we set the bits in the outcode?
How do you suppose we use them?
xmin xmax
0000 00100001
1001
0101 0100
1000 1010
0110
ymax
ymin
19. Cohen-Sutherland Line Clipping
19
Set bits with simple tests
x > xmax y < ymin etc.
Assign an outcode to each vertex of line
If both outcodes = 0, trivial accept
bitwise AND vertex outcodes together
If result ≠ 0, trivial reject
As those lines lie on one
side of the boundary lines
0000 00100001
1001
0101 0100
1000 1010
0110
ymax
ymin
20. Cohen-Sutherland Line Clipping
20
If line cannot be trivially accepted or rejected,
subdivide so that one or both segments can be
discarded
Pick an edge that the line crosses (how?)
Intersect line with edge (how?)
Discard portion on wrong side of edge and assign
outcode to new vertex
Apply trivial accept/reject tests; repeat if necessary
21. Cohen-Sutherland Line Clipping
21
Outcode tests and line-edge intersects are quite fast
(how fast?)
But some lines require multiple iterations:
Clip top
Clip left
Clip bottom
Clip right
Fundamentally more efficient algorithms:
Cyrus-Beck uses parametric lines
Liang-Barsky optimizes this for upright volumes
23. Clipping Polygons
23
We know how to clip a single line segment
How about a polygon in 2D?
How about in 3D?
Clipping polygons is more complex than clipping the
individual lines
Input: polygon
Output: polygon, or nothing
When can we trivially accept/reject a polygon as
opposed to the line segments that make up the
polygon?
24. Why Is Clipping Hard?
24
What happens to a triangle during clipping?
Possible outcomes:
Triangletriangle Trianglequad Triangle5-gon
How many sides can a clipped triangle have?
28. Sutherland-Hodgman Clipping
28
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
29. Sutherland-Hodgman Clipping
29
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
30. Sutherland-Hodgman Clipping
30
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
31. Sutherland-Hodgman Clipping
31
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
32. Sutherland-Hodgman Clipping
32
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
33. Sutherland-Hodgman Clipping
33
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
34. Sutherland-Hodgman Clipping
34
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
35. Sutherland-Hodgman Clipping
35
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
36. Sutherland-Hodgman Clipping
36
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
37. Sutherland-Hodgman Clipping
37
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
Will this work for non-rectangular clip regions?
What would
3-D clipping
involve?
38. Sutherland-Hodgman Clipping
38
Input/output for algorithm:
Input: list of polygon vertices in order
Output: list of clipped polygon vertices consisting of old
vertices (maybe) and new vertices (maybe)
Note: this is exactly what we expect from the
clipping operation against each edge
This algorithm generalizes to 3-D
Show movie…
39. Sutherland-Hodgman Clipping
39
We need to be able to create clipped polygons from
the original polygons
Sutherland-Hodgman basic routine:
Go around polygon one vertex at a time
Current vertex has position p
Previous vertex had position s, and it has been added to the
output if appropriate
40. Sutherland-Hodgman Clipping
40
Edge from s to p takes one of four cases:
(Purple line can be a line or a plane)
inside outside
s
p
p output
inside outside
s
p
no output
inside outside
s
p
i output
inside outside
sp
i output
p output
41. Sutherland-Hodgman Clipping
41
Four cases:
s inside plane and p inside plane
Add p to output
Note: s has already been added
s inside plane and p outside plane
Find intersection point i
Add i to output
s outside plane and p outside plane
Add nothing
s outside plane and p inside plane
Find intersection point i
Add i to output, followed by p
42. Point-to-Plane test
42
A very general test to determine if a point p is
“inside” a plane P, defined by q and n:
(p - q) • n < 0: p inside P
(p - q) • n = 0: p on P
(p - q) • n > 0: p outside P
P
n
p
q
P
n
p
q
P
n
p
q
43. Point-to-Plane Test
43
Dot product is relatively expensive
3 multiplies
5 additions
1 comparison (to 0, in this case)
Think about how you might optimize or special-case
this
44. Finding Line-Plane Intersections
44
Use parametric definition of edge:
E(t) = s + t(p - s)
If t = 0 then E(t) = s
If t = 1 then E(t) = p
Otherwise, E(t) is part way from s to p
45. Finding Line-Plane Intersections
45
Edge intersects plane P where E(t) is on P
q is a point on P
n is normal to P
(E(t) - q) • n = 0
(s + t(p - s) - q) • n = 0
t = [(q - s) • n] / [(p - s) • n]
The intersection point i = E(t) for this value of t
46. Line-Plane Intersections
46
Note that the length of n doesn’t affect result:
t = [(q - s) • n] / [(p - s) • n]
Again, lots of opportunity for optimization
48. 3-D Clipping
48
Before actually drawing on the screen, we have to
clip (Why?)
Can we transform to screen coordinates first, then
clip in 2D?
Correctness: shouldn’t draw objects behind viewer
What will an object with negative z coordinates do in our
perspective matrix?
49. Recap: Perspective Projection Matrix
49
Example:
Or, in 3-D coordinates:
Multiplying by the projection matrix gets us the 3-D
coordinates
The act of dividing x and y by z/d is called the
homogeneous divide
=
10100
0100
0010
0001
z
y
x
ddz
z
y
x
d
dz
y
dz
x
,,
50. Clipping Under Perspective
50
Problem: after multiplying by a perspective matrix
and performing the homogeneous divide, a point at
(-8, -2, -10) looks the same as a point at (8, 2, 10).
Solution A: clip before multiplying the point by the
projection matrix
I.e., clip in camera coordinates
Solution B: clip after the projection matrix but
before the homogeneous divide
I.e., clip in homogeneous screen coordinates
51. Clipping Under Perspective
51
We will talk first about solution A:
Clip against
view volume
Apply projection
matrix and
homogeneous
divide
Transform into
viewport for
2-D display
3-D world
coordinate
primitives
Clipped world
coordinates
2-D device
coordinates
Canonical screen
coordinates
53. Perspective Projection
53
The viewing frustum consists of six planes
The Sutherland-Hodgeman algorithm (clipping
polygons to a region one plane at a time) generalizes
to 3-D
Clip polygons against six planes of view frustum
So what’s the problem?
The problem: clipping a line segment to an arbitrary
plane is relatively expensive
Dot products and such
54. Perspective Projection
54
In fact, for simplicity we prefer to use the canonical
view frustum:
x or y
1
-1
z
-1
Front or
hither plane
Back or yon plane
Why is this going to be
simpler?
Why is the yon plane
at z = -1, not z = 1?
55. Clipping Under Perspective
55
So we have to refine our pipeline model:
Note that this model forces us to separate projection
from modeling & viewing transforms
Apply
normalizing
transformation
projection
matrix;
homogeneous
divide
Transform into
viewport for
2-D display
3-D world
coordinate
primitives
2-D device
coordinates
Clip against
canonical
view
volume
56. Clipping Homogeneous Coords
56
Another option is to clip the homogeneous
coordinates directly.
This allows us to clip after perspective projection:
What are the advantages?
Clip
against
view
volume
Apply
projection
matrix
Transform into
viewport for
2-D display
3-D world
coordinate
primitives
2-D device
coordinates
Homogeneous
divide
57. Clipping Homogeneous Coords
57
Other advantages:
Can transform the canonical view volume for perspective
projections to the canonical view volume for parallel
projections
Clip in the latter (only works in homogeneous coords)
Allows an optimized (hardware) implementation
Some primitives will have w ≠ 1
For example, polygons that result from tesselating splines
Without clipping in homogeneous coords, must perform divide
twice on such primitives
58. Clipping Homogeneous Coords
58
So how do we clip homogeneous coordinates?
Briefly, thus:
Remember that we have applied a transform to normalized
device coordinates
x, y [-1, 1]
z [0, 1]
When clipping to (say) right side of the screen (x = 1), instead
clip to (x = w)
Can find details in book or on web
59. Clipping: The Real World
59
In some renderers, a common shortcut used to be:
But in today’s hardware, everybody just clips in
homogeneous coordinates
Projection
matrix;
homogeneous
divide
Clip in 2-D
screen
coordinates
Clip against
hither and
yon planes
Transform into
screen
coordinates
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