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Basic theory of curve and surface

Basic theory of curve and surface. Geometric representation. Parametric Non-parametric Explicit Implicit. x = x(u), y = y(u). y = f(x). f(x, y) = 0. Geometric representation. Example - circle Parametric Non-parametric Explicit Implicit. x = R cos  , y = R sin .

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Basic theory of curve and surface

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  1. Basic theory of curve and surface Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  2. Geometric representation • Parametric • Non-parametric • Explicit • Implicit x = x(u), y = y(u) • y = f(x) • f(x, y) = 0 Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  3. Geometric representation • Example - circle • Parametric • Non-parametric • Explicit • Implicit x = R cos, y = R sin • y = R2 – x2 • x2 + y2 – R2 = 0 Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  4. Geometric representation • Each form has its own advantages and disadvantages, depending on the application for which the equation is used. Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  5. Non-parametric (explicit) • y = f(x) • Only one y value for each x value • Cannot represent closed or multiple-valued curves such as circle Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  6. Non-parametric (implicit) f(x,y) = 0 ax2 + bxy + cy2 + dx + ey + f = 0 • Advantages – can produce several type of curve – set the coefficients • Disadvantages • Not sure which variable to choose as the independent variable Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  7. Non-parametric (cont) • Disadvantages • Non-parametric elements are axis dependant, so the choice of coordinate system affects the ease of using the element and calculating their properties. • Problem  if the curve has a vertical slope (infinity). • They represent unbounded geometry e.g ax + by + c = 0 define an infinite line Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  8. parametric • Express relationship for the x, y and z coordinates not in term of each other but of one or more independent variable (parameter). • Advantages • Offer more degrees of freedom for controlling the shape • (non-parametric) y= ax3 + bx2 + cx + d • (parametric) x = au3 + bu2 + cu + d • y = eu3 + fu2 + gu + h Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  9. Parametric (cont) • Advantages (cont) • Transformations can be performed directly on parametric equations. • Parametric forms readily handle infinite slopes without breaking down computationally dy/dx = (dy/du)/ (dx/du) • Completely separate the roles of the dependent and independent variable. Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  10. Parametric (cont) • Advantages (cont) • easy to express in the form of vectors and matrices • Inherently bounded. Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  11. t3 t2 t4 t5 t6 t1 Parametric curve • Use parameter to relate coordinate x and y (2D). • Analogy • Parameter t (time) – [ x(t), y(t) as the position of the particle at time t ] y x Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  12. line b a b a ray b a Line segment Parametric curve • Fundamental geometric objects – lines, rays and line segment All share the same parametric representation Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  13. b a Parametric line a = (ax, ay), b = (bx, by) x(t ) = ax + (bx - ax)t y(t) = ay + (by - ay)t • Parameter t is varied from 0 to 1 to define all point along the line • When t = 0, the point is at “a”, as t increases toward 1, the point moves in a straight line to b. • For line segment : 0  t  1 • For line : -  t   • For ray : 0  t   Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  14. Parametric line • Example • A line from point (2, 3) to point (-1, 5) can be represented in parametric form as x(t) = 2 + (-1 – 2)t = 2 – 3t y(t) = 3 + (5 – 2)t = 3 + 3t Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  15. Parametric line • Positions along the line are based upon the parameter value • E.g midpoint of a line occurs at t = 0.5 • Exercise : Find the parametric form for the segment with endpoints (2, 4, 1) and (7, 5, 5). Find the midpoint of the segment by using t = 0.5 Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  16. Parametric line • Answer: Parametric form: x(t) = 2 + (7 –2)t = 2 + 5t y(t) = 4 + (5 – 4)t = 4 + t z(t) = 1 + (5 – 1)t = 1 + 4t Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  17. Parametric line • Answer Midpoint x(0.5) = 2 + 5(0.5) = 5.5  6 Y(0.5) = 4 + (0.5) = 4.5  5 Z(0.5) = 1 + 4(0.5) = 3  3 Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  18. Parametric curve (conic section) • Another basic example • Conic section - the curves / portions of the curves, obtained by cutting a cone with a plane. • The section curve may be a circle, ellipse, parabola or hyperbola. hyperbola parabola ellipse Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  19. Parametric curve (circle) • The simplest non-linear curve - circle - circle with radius R centered at the origin • x(t) = R cos(2t) • y(t) = R sin(2t) 0  t  1 Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  20. Circular arc Parametric curve (circle) • If t = 0.125  a 1/8 circle • t = 0.25  a 1/4 circle • t = 0.5  a ½ circle t = 1  a circle Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  21. Parametric curve (circle) • Circle with center at (xc, yc) • x(t) = R cos(2t) + xc, • y(t) = R sin(2t) + yc , Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  22. Parametric curve • Ellipse • x(t) = a cos(2t) • y(t) = b sin(2t) • Hyperbola • x(t) = a sec(t) • y(t) = b tan(t) • parabola • x(t) = at2 • y(t) = 2at b a b a Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  23. Control for this curve • Shape (based upon parametric equation) • Location (based upon center point) • Size • Arc (based upon parameter range) • Radius (a coefficient to unit value) Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  24. Parametric curve • Generally • A parametric curve in 3D space has the following form • F: [0, 1] (x(t), y(t), z(t)) • where x(), y() and z() are three real-valued functions. Thus, F(t) maps a real value t in the closed interval [0,1] to a point in space • for simplicity, we restrict the domain to [0,1]. Thus, for each t in [0,1], there corresponds to a point (x(t), y(t), z(t) ) in space. If z( ) is removed - ? A curve in a coordinate plane Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  25. Tangent vector and tangent line • Tangent vector • Vector tangent to the slope of curve at a given point • Tangent line • The line that contains the tangent vector Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  26. Compute tangent vector • F(t) = (x(t), y(t), z(t)) • Tangent vector : • F’(t) = (x’(t), y’(t), z’(t)) Where x’(t)= dx/dt, y’(t)= dy/dt, z’(t)= dz/dt • Magnitude /length • If vector V = (a, b, c)  |V| =  a2 + b2 + c2 • Unit vector • Uv = V / |V| Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  27. Compute tangent line • Tangent line at t is either • F(t) + uF’(t) or • F(t) + u(F’(t)/|F’(t)|)  if prefer unit vector • u is a parameter for line Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  28. example • Question: - given a Circle, F(t) = (Rcos(2t), R sin(2t)) , 0  t  1 Find tangent vector at t and tangent line at F(t). Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  29. example • Answer dx = Rcos(2t), dy = R sin(2t) x’(t) = dx/dt = - 2 Rsin (2t), y’(t) = dy/dt = 2Rcos(2t) Tangent vector = (- 2 Rsin (2t), 2Rcos(2t)) Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  30. example • Answer • Tangent line • F(t) + u(F’(t)) • (Rcos(2t), R sin(2t)) + u(- 2 Rsin (2t), 2Rcos(2t)) • (Rcos(2t) + u(- 2 Rsin(2t))) , (R sin(2t) + u(2Rcos(2t))) Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  31. Example • Check / prove • Let say, t = 0, Tangent vector = (- 2 Rsin (2t), 2Rcos(2t)) = (0, 2R) tangent line = (Rcos(2t) + u(- 2 Rsin(2t))) , (R sin(2t) + u(2Rcos(2t))) = (R, u(2R)) R Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  32. Tangent vector • Slope of the curve at any point can be obtained from tangent vector. • Tangent vector at t = (x’(t), y’(t)) • Slope at t = dy/dx = y’(t)/x’(t) Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  33. curvature • The curvature at a point measures the rate of curving (bending) as the point moves along the curve with unit speed • When orientation is changed the curvature changes its sign, the curvature vector remains the same • Straight line  curvature = ? Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  34. curvature P P • Circle is tangent to the curve at P • lies toward the concave or inner side of the curve at P • Curvature = 1/r , r radius Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  35. curvature • The curvature at u, k(u), can be computed as follows: • k(u) = | f'(u) × f''(u) | / | f'(u) |3 • How about curvature of a circle ? Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  36. Curve use in design • Engineering design requires ability to express complex curve shapes (beyond conic) and interactive. • Bounding curves for turbine blades, ship hulls, etc • Curve of intersection between surfaces. Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  37. Curve use in design • A design is “GOOD” if it meets its design specifications : These may be either : • Functional - does it works. • Technical - is it efficient, does it meet certain benchmark or standard. • Aesthetic - does it look right, this is both subjective and opinion is likely to change in time or combination of both. Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  38. Representing complex curves • Typically represented • A series of simpler curve (each defined by a single equation) pieced together at their endpoints.(piecewise construction) Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  39. Representing complex curves • Typically represented • Simple curve may be linear or polynomial • Equation for simpler curves based on control points (data points used to define the curve). Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  40. An interactive curve design a) Desired curve b) User places points c) The algorithm generates many points along a “nearby” curve Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

  41. An interactive curve design • Interactive design consists of the following steps • Lay down the initial control points • Use the algorithm to generate the curve • If the curve satisfactory, stop. • Adjust some control points • Go to step 2. Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

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