Parametric Line equations

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Presentation transcript:

Parametric Line equations (x2,y2) For a line from (x1,y1) to (x2,y2) x = x1 + (x2 – x1)*u = x1(1-u) + x2u y = y1(1-u) + y2u Where 0 ≤ u ≤1 (x1,y1) CS-321 Dr. Mark L. Hornick

Vector form of Parametric equation p2 =(x2,y2) x = x1(1-u) + x2u y = y1(1-u) + y2u Or P(u) = p1(1-u) + p2u Where P = {x y} p1 =(x1,y1) CS-321 Dr. Mark L. Hornick

A Bézier Curve (of order 2) CS-321 1/15/2019 A Bézier Curve (of order 2) P1 P0 P2 CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Bézier Curves (or Bezier Splines) CS-321 1/15/2019 Bézier Curves (or Bezier Splines) Most common curves in C.G. Uses control points p1, p2, … Determine boundaries of curve shape Curve does not pass through them Except at beginning and end Bezier Polynomial function Degree = # control points – 1 Continuity through all derivatives CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

General form of Bézier Curve CS-321 1/15/2019 General form of Bézier Curve n+1 control points: 0..n; u = 0 at p0, u=1 at pn Bernstein Polynomials exhibit the property that, for any value of u: CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Binomial Coefficients CS-321 1/15/2019 Binomial Coefficients k n 0 1 2 3 4 5 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Bézier Curve: n = 1 P1 P0 CS-321 1/15/2019 Dr. Mark L. Hornick

Bézier Curve: n = 2 P1 P0 P2 CS-321 1/15/2019 Dr. Mark L. Hornick

Bernstein Polynomial: n = 2 CS-321 1/15/2019 Bernstein Polynomial: n = 2 CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Bernstein Polynomial: n = 3 CS-321 1/15/2019 Bernstein Polynomial: n = 3 0.2 0.4 0.6 0.8 1 0.5 1.5 u P0 P1 P2 P3 CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Bézier Summary Curve shape Smooth curve CS-321 1/15/2019 Bézier Summary Curve shape Influenced by all control points Change one point – whole curve changes shape Influence increased near each point Contained in convex hull of control points Smooth curve Polynomial degree = #points – 1 Parametric continuity through all derivatives Does not pass through all points CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Curved Lines and Surfaces CS-321 1/15/2019 Curved Lines and Surfaces Problem: How to model arbitrarily curved surfaces Boundary representation approaches approximate with linear/polygon mesh …or with curved surface patches CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Spline Surface Two sets of orthogonal spline curves CS-321 1/15/2019 CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick