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Chap. 14 Curves Mathematics for Computer Graphics Applications

Chap. 14 Curves Mathematics for Computer Graphics Applications. Seminar for Beginner Summer 2002 Jang Su-Mi 2002-08-07. Parametric Equations of Curve. x = x(u) y = y(u) z = z(u) x(u) = au 2 + bu + c p = p (u) p (u) = [x(u) y(u) z(u)]. Plane Curves(1).

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Chap. 14 Curves Mathematics for Computer Graphics Applications

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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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