Basic Curve Surface
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Basic Curve Surface

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Basic Curve Surface Basic Curve Surface Presentation Transcript

  • Basic theory of curve and surface
  • Geometric representation
    • Parametric
    • Non-parametric
      • Explicit
      • Implicit
      • y = f(x)
      • f(x, y) = 0
    x = x(u), y = y(u)
  • Geometric representation
    • Example - circle
    • Parametric
    • Non-parametric
      • Explicit
      • Implicit
      • y =  R 2 – x 2
      • x 2 + y 2 – R 2 = 0
    x = R cos  , y = R sin 
    • Each form has its own advantages and disadvantages, depending on the application for which the equation is used.
    Geometric representation
  • Non-parametric (explicit)
    • Only one y value for each x value
    • Cannot represent closed or multiple-valued curves such as circle
      • y = f(x)
  • Non-parametric (implicit)
    • Advantages – can produce several type of curve – set the coefficients
    • Disadvantages
      • Not sure which variable to choose as the independent variable
    f(x,y) = 0 ax 2 + bxy + cy 2 + dx + ey + f = 0
  • 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
  • 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= ax 3 + bx 2 + cx + d
        • (parametric) x = au 3 + bu 2 + cu + d
        • y = eu 3 + fu 2 + gu + h
  • 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.
  • Parametric (cont)
    • Advantages (cont)
      • easy to express in the form of vectors and matrices
      • Inherently bounded.
  • 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 ]
    x y t1 t2 t3 t4 t5 t6
  • Parametric curve
    • Fundamental geometric objects – lines, rays and line segment
    All share the same parametric representation a b line a b ray a b Line segment
  • Parametric line
    • a = (a x , a y ), b = (b x , b y )
    • x(t ) = a x + (b x - a x )t
    • y(t) = a y + (b y - a y )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  
    a b
  • 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
  • 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
  • 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
    • 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
    Parametric line
    • 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.
    Parametric curve (conic section) ellipse hyperbola parabola
  • 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
    • If t = 0.125  a 1/8 circle
    Parametric curve (circle)
      • t = 0.25  a 1/4 circle
      • t = 0.5  a ½ circle
    t = 1  a circle Circular arc
    • Circle with center at (x c , y c )
    • x(t) = R cos(2  t) + x c ,
    • y(t) = R sin(2  t) + y c ,
    Parametric curve (circle)
  • 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) = at 2
      • y(t) = 2at
    a b a b
  • 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)
  • 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
  • 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
    • 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| =  a 2 + b 2 + c 2
    • Unit vector
      • Uv = V / |V|
    Compute tangent vector
  • Compute tangent line
    • Tangent line at t is either
      • F(t) + u F’(t)
      • or
      • F(t) + u (F’(t)/|F’(t)|)  if prefer unit vector
      • u is a parameter for line
  • 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).
  • 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))
  • 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)))
  • 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
  • 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)
    • 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 = ?
    curvature
  • curvature
    • 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
    P P
  • 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 ?
  • 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.
    • 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.
    Curve use in design
  • Representing complex curves
    • Typically represented
      • A series of simpler curve ( each defined by a single equation ) pieced together at their endpoints.( piecewise construction )
  • 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 ).
  • An interactive curve design a) Desired curve b) User places points c) The algorithm generates many points along a “nearby” curve
    • 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.
    An interactive curve design