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Introduction to Parametric Equations and Curve Mathematics for Computer Graphics

This seminar introduces fundamental concepts of parametric equations and their applications in computer graphics. Participants will learn how to express curves using algebraic and geometric forms, understand the relationship between control points and curve parameters, and explore blending functions for smooth transitions between curves. Key topics include matrix algebra, the tangent vector calculations, and continuity types (parametric and geometric). This beginner-friendly session will equip participants with the essential tools to create and manipulate curves in various graphic applications.

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Introduction to Parametric Equations and Curve Mathematics for Computer Graphics

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  1. Chap. 14 CurvesMathematics for Computer Graphics Applications Seminar for Beginner Summer 2002 Jang Su-Mi 2002-08-07

  2. Parametric Equations of Curve x = x(u) y = y(u) z = z(u) x(u) = au2 + bu + c p = p(u) p(u) = [x(u) y(u) z(u)]

  3. Plane Curves(1) x(u) = axu2 + bxu + cx y(u) = ayu2 + byu + cy z(u) = azu2 + bzu + cz p(u) = au2 + bu + c  Algebraic form

  4. Plane Curve(2) 3 Pointare needed. p0=[x0 y0 z0] ; u = 0 p0.5=[x0.5 y0.5 z0.5] ; u = 0.5 p1=[x1 y1 z1] ; u = 1  Algebraic form에 대입 x0 = cx x0.5 = 0.25ax + 0.5bx + cx x1 = ax + bx + cx  y, z에 대해서도 비슷한 결과

  5. Plane Curve(3) ax = 2x0 - 4x0.5 + 2x1 bx = -3x0 +x0.5 - x1 cx = x0 ax bx cx에 대하여 푼 것 x(u) = (2x0 - 4x0.5 + 2x1)u2 +(-3x0 +x0.5 - x1)u + x0 y(u), z(u)도 비슷한 결과 x(u) = (2u2 – 3u +1)x0 + (-4u2 + 4u) x0.5 +(2u2 – u) x1 • x0 x0.5 x1에 대하여 정리 p(u) = (2u2-3u+1)p0+(-4u2+4u)p0.5+(2u2–u)p1  Geometric form

  6. Plane Curves(4) • Matrix Algebra (Algebraic form) p(u) = au2 + bu + c  a [u2 u 1] b = au2 + bu + c c U = [u2 u 1] A = [a b c]T = ax ay az  p(u) = UA bx by bz cx cy cz Algebraic coefficients

  7. Space Curves(5) • Matrix Algebra (Geometric form) p(u) = (2u2-3u+1)p0+(-4u2+4u)p0.5+(2u2–u)p1  p(u) =[(2u2-3u+1) (-4u2+4u) (2u2–u)] [p0 p0.5 p1]T F = [(2u2-3u+1) (-4u2+4u) (2u2–u)] P = [p0 p0.5 p1]T = x0 y0 z0 x0.5 y0.5 z0.5 x1 y1 z1 p(u)=FP Blending function matrix Control Point matrix Geometric Coefficients

  8. Plane Curves(6) FP = UA F = [u2 u 1] 2 -4 2 -3 4 -1 M 1 0 0 F = UM UMP = UA MP = A A = MP P = M-1A Basis transformation matrix

  9. Space Curve • Cubic Polynomials : x(u) y(u) z(u), p(u) • 4 Points are needed : p0 p1/3 p2/3 p1 • Same process with the Plane curve p(u) = UA Algebraic form p(u) = GP Geometric form G = UN N : basis transformation matrix GP = UA UNP = UA A = NP

  10. The Tangent Vector • Use 2 end point, 2 tangents instead of 4 point. (p0 p1 pu0 pu1 ) • Tangent vector pu(u) = [ dx(u)i/dudy(u)j/du dz(u)k/du] pu = [xu yu zu] x(u) = axu3 + bxu2 + cxu + dx xu = 3axu2 + 2bxu + cx

  11. The Tangent Vector u=0, u=1대입  x0 x1 xu0 xu1에 대하여 정리 • ax bx cx dx 에 대하여 정리  치환대입 정리 x(u) = (2x0-2x1+xu0 +xu1 )u3+(-3x0 +3x1-2xu0-xu1 ) u2+xu0u +x0 • x0 x1 xu0 xu1에 대하여 정리 x(u) = (2u3-3u2+1)x0 +(-2u3 +3u2) x1 +(u3 -2u2 +u)xu0 +(u3-u2)xu1 p(u) = (2u3-3u2+1)p0 +(-2u3 +3u2)p1 +(u3 -2u2 +u)pu0 +(u3-u2)pu1 F B

  12. The Tangent Vector p(u) = UA p(u) = FB F = UM UMB = UA A = MB (magnitude of the tangent vector account into) pu0 = m0t0 pu1 = m1t1 p(u) = (2u3-3u2+1)p0 +(-2u3 +3u2)p1 +(u3 -2u2 +u)m0t0 +(u3-u2)m1t1

  13. F blending Function G blending Function Blending Function

  14. Reparameterization • reverse direction

  15. Continuity and Composit Curves • Parametric Continuity : Cn • Geometric Continuity : Gn

  16. Approximating a Conic Curve • Conic Curves • Hyperbola • Parabola • Ellipse

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