# Computer Graphics Spline Computer Graphics-Books Pdf

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Computer Graphics Computer Graphics, The basic one. Cubic B Splines, Uniform Cubic B Splines, Cubic B splines with uniform knot vector is the The unweighted spline set of 10 control points 10 splines. 14 knots and but only 7 intervals, most commonly used form of B splines You can see why t3 to t4 is the first interval with a curve. X t t T MQ i for ti t t i 1 since it is the first with all four B Spline functions. where Q i xi 3 xi t9 to t10 is the last interval, t T t ti 3 t ti 2 t ti 1 0 6 8 m t. 10 10 2008, i 3 i Lecture 5 7 10 10 2008 3 4 Lecture 5 8.
Computer Graphics Computer Graphics, Cubic Uniform B Spline. Domain of the function, 2D example, Order k Degree k 1 For each i 4 there is a knot between Qi 1 and Qi at t ti. Initial points at t3 and tm 1 are also knots The following. Control points Pi i 0 m illustrates an example with control points set P0 P9. Knots tj j 0 k m, The domain of the function tk 1 t tm 1 Control point. Below k 4 m 9 domain t3 t t10, 3 m 1 10 10 2008 Lecture 5 10. Computer Graphics Computer Graphics, Uniform Non rational B Splines Uniform Non rational B Splines.
First segment Q3 is defined by point P0 through P3 over the Second segment Q4 is defined by point P1 through P4 over. range t3 0 to t4 1 So m at least 3 for cubic spline the range t4 1 to t5 2. P1 Control point, P1 Control point, 10 10 2008 Lecture 5 11 10 10 2008 Lecture 5 12. Computer Graphics Computer Graphics, The shape of the basis functions. A more general definition, A Bspline of order k is a parametric curve composed of a linear Bi 2 linear basis functions. combination of basis B splines Bi n, Order 2 degree 1. Pi i 0 m the control points, m C0 continuous, Knots tj j 0 k m p t Pi Bi n t.
The B splines can be defined by, 1 t t t i 1, 0 otherwise. Bi k t Bi k 1 t i k Bi 1 k 1 t, ti k Lecture ti k 1 ti. 10 10 2008 13 10 10 2008 http www ibiblio org e notes Splines Basis htm. Lecture 5 14, Computer Graphics Computer Graphics, The shape of the basis functions The shape of the basis functions. Bi 3 Quadratic basis functions Bi 4 Cubic basis functions. Order 3 degree 2 Order 4 degree 3, C1 continuous C2 continuous. 10 10 2008 http www ibiblio org e notes Splines Basis htm. Lecture 5 15 10 10 2008 http www ibiblio org e notes Splines Basis htm. Lecture 5 16, Computer Graphics Computer Graphics, Uniform non uniform B splines Non uniform B splines.
Uniform B splines Blending functions no longer the same for each interval. Advantages, The knots are equidistant non equidistant. Continuity at selected control points can be reduced to. The previous examples were uniform B splines C1 or lower allows us to interpolate a control point. without side effects, t0 t1 t2 t m were equidistant same interval Can interpolate start and end points. Easy to add extra knots and control points, Parametric interval between knots does not Good for shape modelling. have to be equal, Non uniform B splines, 10 10 2008. Computer Graphics Computer Graphics, Controlling the shape through.
Controlling the shape of the curves, control points. Can control the shape through P2, Control points, Overlapping the control points to make it pass First knot shown with 4. through a specific point control points and their, Knots convex hull. Changing the continuity by increasing the, multiplicity at some knot non uniform bsplines. 10 10 2008 Lecture 5 20, Computer Graphics Computer Graphics.
Controlling the shape through, Repeated control point. control points, First two curve segments, First two curve segments shown with their respective. shown with their respective convex hulls, convex hulls. The curve is forced to lie on, Centre Knot must lie in the the line that joins the 2. intersection of the 2 convex convex hulls, P1 P3 P1 P2 P4.
10 10 2008 Lecture 5 21 10 10 2008 Lecture 5 22, Computer Graphics Computer Graphics. Triple control point Controlling the shape through knots. First two curve segments, P0 shown with their respective Smoothness increases with order k in Bi k. convex hulls Quadratic k 3 gives up to C1 continuity. Cubic k 4 gives up to C2 continuity, Both convex hulls collapse. to straight lines all the However we can lower continuity order too with Multiple. curve must lie on these lines Knots ie ti ti 1 ti 2 Knots are coincident and so. now we have non uniform knot intervals, A knot with multiplicity p is continuous to the. k 1 p th derivative, A knot with multiplicity k has no continuity at all i e the.
P1 P2 P3 curve is broken at that knot B t 1 t t t, 0 otherwise. 10 10 2008 Lecture 5 23 10 10 2008 LectureB5i k t Bi k 1 t i k Bi 1 k 1 t 24. t i k 1 ti ti k 1 ti, Computer Graphics Computer Graphics. B Splines at multiple knots Knot multiplicity, Cubic B spline Consider the uniform cubic n 4 B spline curve. Multiplicities are indicated t 0 1 13 m 9 n 4 7 segments. Computer Graphics Knot multiplicity Computer Graphics Knot multiplicity. Double knot at 5, knot 0 1 2 3 4 5 5 6 7 8 9 10 11 12. Triple knot at 5, 6 segments continuity 1, knot 0 1 2 3 4.
5 5 5 6 7 8 9 10, 5 segments, Computer Graphics Computer Graphics. Knot multiplicity Summary of B Splines, Quadruple knot at 5 Functions that can be manipulated by a series of control. points with C2 continuity and local control, 4 segments Don t pass through their control points although can be. Knots are equally spaced in t, Non Uniform, Knots are unequally spaced. Allows addition of extra control points anywhere in the set. 10 10 2008 Lecture 5 30, Computer Graphics Computer Graphics.
Summary cont 2nd Practical, Do not have to worry about the continuity at Use OpenGL to draw the teapot. the join points, You must extend your code in the first. For interactive curve modelling assignment, B Splines are very good. Bonus marks for making it nice, Bump mapping, Texture mapping. Or whatever, Deadline 10th December, 10 10 2008 Lecture 5 31.
Computer Graphics, Reading for this lecture, Foley at al Chapter 11 sections 11 2 3. 11 2 4 11 2 9 11 2 10 11 3 and 11 5, Introductory text Chapter 9 sections 9 2 4. 9 2 5 9 2 7 9 2 8 and 9 3, 10 10 2008 Lecture 5 33. Computer Graphics 10 10 2008 Lecture 5 2 Spline A long flexible strips of metal used by draftspersons to lay out the surfaces of airplanes cars and ships Ducks weights attached to the splines were used to pull the spline in different directions The metal splines had second order continuity Computer Graphics 10 10 2008 Lecture 5 3 B Splines for basis splines B Splines

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