Parametric curves curves and surfaces can have explicit, implicit. Cubic curves are commonly used in graphics because. Curves and surface we introduce three ways to model curves and surfaces, paying most attention to the parametric polynomial forms. Towards g 2 curve design with timmer parametric cubic.
A curve will either interpolate or approximate control points. Curves and surfaces 15462 computer graphics i lecture 9 02192002 15462 graphics i 2 goals. A cubic polynomial t ranging from 0 to 1 polynomial can be speci. Results in a smooth parametric curve p t just means that we specify xt and yt in practice. A curve tutorial for introductory computer graphics. An introduction to splines for use in computer graphics and geometric modeling, 1987.
Curves and surfacescurves and surfaces 15462 computer graphics i lecture 9. Parametric curve in computer graphics in hindi parametric cubic curve in computer graphics parametric equations are commonly used to express the coordinates of the points that make up a. Lecture outline more on curves and parametric bicubic. An adaptive forward differencing algorithm is presented tot rapid rendering of cubic curves and bicubic surfaces. A twodimensional parametric curve has the following form. Designing parametric cubic curves computer science. You can see why t 3 to t 4 is the first interval with a curve since it is the first with all four b. Computer graphics curves and surfaces hermitebezier curves, bsplines, and nurbs by ulf assarsson. Two of the four conditions required to specify a cubic are taken by placement of control points. Here you will find the syllabus of first subject in bca semesteriv th, which is computer graphics and multimedia application the course code of computer graphics and multimedia application is bca. University of texas at austin cs384g computer graphics fall 2010 don fussell 11 generalizing from parametric curves flashback to curves. Cubic piecewise parametric polynomial curves from now on, consider cubic blending functions o all ideas generalize to higher degrees in cagd, higherorder functions are often used o hard to control wiggles in graphics, piecewise cubic curves will do o smallest degree that allows c2 continuity for arbitrary curves. We also discuss how curves and surfaces can be rendered on current graphics.
As the field of computer graphics develops, techniques for modeling complex curves and surfaces are increasingly important. Cubic polynomials are very popular in curve design. Computer graphics and interaction dh2323 spring 2015 p4 bezier curves, splines and surfaces. Parametric representations are the most common in computer graphics. A geometric characterization of parametric cubic curves. Consider the following parameterizations for a line. Behzeeay is a parametric curve used in computer graphics and related fields. Explicit or parametric curve descriptions provide a mapping from a free. Computer graphics hermite curves, bsplines and nurbs week 2, lecture 4 david breen, william regli and maxim peysakhov department of computer science drexel university additional slides from don fussell, university of texas 2 outline hermite curves more types of curves. Parametric curves department of computer science pre.
Note that many parametric curves might have the same implicit form. Bezier suggested using the same 4 data points as with the cubic interpolating curve. Between parametric and implicit curves and surfaces christoph m. Then with most graphics systems one must approximate the curve. Set up equations for cubic parametric curve recall. Designing parametric cubic curves unm computer science. Computer graphics curves and surfaces hermitebezier curves, bsplines, and nurbs by ulf assarsson most of the material is originally made by edward angel and is. A curve tutorial for introductory computer graphics michael gleicher department of computer sciences university of wisconsin, madison october 14, 2004 1 curves mathematically, a curve is. This is because parametric curves represent a particular parametrization of the underlying algebraic curve. Parametric curves curves and surfaces can have explicit, implicit, and parametric representations. We directly defined parametric function fu, as a cubic polynomial. Parametric cubic splines are the curves of choice for many applications of computer graphics. Curves can be broadly classified into three categories.
Arclength parameterized spline curves for realtime. Carpenter, eece ksu november 22, 1999 revised november 20, 2001, april 24, 2002, april 14, 2004 1 introduction suppose one wishes to draw a curve in three dimensions. A geometric characterization of parametric cubic curves l 151 point must be constructed from the control points, and since the diagram has a fairly large number of disconnected regions. Pdf towards g 2 curve design with timmer parametric cubic. A longstanding problem in computer graphics is to find a planar curve that is shaped the way you want it. Thus, if a nondegenerate parametric cubic curve has a cusp, then it cannot have a loop or an inflection point. Parametric curves cse 457 winter 2014 2 reading required.
Computer graphics curves in computer graphics, we often need to draw different. University of texas at austin cs384g computer graphics fall 2010 don fussell 8 cubic bezier curves specifying tangent vectors at endpoints isnt always convenient for geometric modeling we may prefer making all the geometric coefficients points, lets call them control points, and label them p 0, p 1, p 2, and p 3. Curves having parametric form are called parametric curves. Parametric representation of curves computer science. We also discuss how curves and surfaces can be rendered on current graphics systems, a process that usually involves subdividing the curved objects into collections of flat primitives. The diagrams produced by our method do not require the construction of an additional characteristic point. Parametric bicubic surfaces cubic polynomials are also used for piecewise interpolation of sets of points on a surface. In graphics and cad, we usually dont have derivative data. A major technique is the use of parametric splines in which a curve is defined by piecing together a succession of curve. A continuous map from a onedimensional space to an ndimensional space. Restrict xu,v, yu,v, zu,v to be polynomial of fixed degree n each cik is a 3element column vector restrict to simple case where 0 w u,v w 1. Parametric curve in computer graphics in hindi parametric cubic curve in computer graphics parametric equations are commonly used to express the coordinates of the points that.
Given a collection of points, the convex hull is the smallest convex set that contains the points. The other two conditions are open to adjust the shape of the curve. An introduction to interpolation and splines kenneth h. Implicit curves implicit curve representations define the set of points on a curve by employing a procedure that can test to see if a point in on the curve. This method adjusts the forward difference step size so that approximately one pixel is generated along an ordinary or. The explicit and implicit curve representations can be used only when the function is known. Curves and surfaces carnegie mellon school of computer. Resolution independent curve rendering using programmable. Computer graphics notesparametric curves and surfaces. Cubic parametric polynomials nml3, gives balance between ease. Cubic polynomials are popular in graphics see section 5. Computer graphics terminology cubic curve representations polygon meshes. Restrict xu, yu, zu to be polynomial in u fix degree n each ck is a column vector. Parametric surfaces department of computer science.
Restrict xu,v, yu,v, zu,v to be polynomial of fixed. A convex set is a collection of points in which the line connecting any pair of points in the set lies entirely within the set. Control points allow us to shapedefine curves visually. Introduction to parametric interpolation for computer. Computer graphics, volume 23, number 3, july 1989 rendering cubic curves and surfaces with integer adaptive forward differencing.
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