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Essential Geometry

Essential Geometry. Equations. Implicit F( x,y ) = 0 Parametric P(t) = x Q(t) = y 0 <= t <= 1. Geometric Operations. Bounding box Efficiency Distance from a point Selection Nearest Point Snapping Inside/Outside Selection. Geometric Operations. R. Cx,Cy. Geometric Operations.

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Essential Geometry

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  1. Essential Geometry

  2. Equations • Implicit • F(x,y) = 0 • Parametric • P(t) = x • Q(t) = y • 0 <= t <= 1

  3. Geometric Operations • Bounding box • Efficiency • Distance from a point • Selection • Nearest Point • Snapping • Inside/Outside • Selection

  4. Geometric Operations R Cx,Cy

  5. Geometric Operations Top Left Right Bottom

  6. Geometric Operations B A Cx,Cy

  7. Geometric Operations

  8. Geometric Operations

  9. Piecewise Continuity • C(0) – same end point • C(1) – same slope at the shared endpoint (Smooth joint) • C(2) – Smooth second derivative at the joint

  10. Parametric form • P(t) = at3 + bt2 + ct + d = x • Q(t) = et3 + ft2+ gt+ h = y

  11. Bezier A2 A3 A0 A1

  12. Bezier B2 A2 B3 B0 A3 A0 B1 A1

  13. Bezier Convex Hull A2 A3 A0 A1

  14. B-Spline A4 A3 A5 A0 A2 A1

  15. Catmull-Rom A4 A3 A5 A0 A1 A2

  16. Matrix Forms

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