Basic theory of curve and surface
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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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Basic theory of curve and surface

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Geometric representation

  • Parametric

  • Non-parametric

    • Explicit

    • Implicit

x = x(u),y = y(u)

  • y = f(x)

  • f(x, y) = 0

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Geometric representation

  • Example - circle

  • Parametric

  • Non-parametric

    • Explicit

    • Implicit

x = R cos,y = R sin

  • y = R2 – x2

  • x2 + y2 – R2 = 0

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Geometric representation

  • Each form has its own advantages and disadvantages, depending on the application for which the equation is used.

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Non-parametric (explicit)

  • y = f(x)

  • Only one y value for each x value

  • Cannot represent closed or multiple-valued curves such as circle

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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

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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

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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

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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.

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Parametric (cont)

  • Advantages (cont)

    • easy to express in the form of vectors and matrices

    • Inherently bounded.

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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

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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

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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  

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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

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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

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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

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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

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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

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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

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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

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Parametric curve (circle)

  • Circle with center at (xc, yc)

  • x(t) = R cos(2t) + xc,

  • y(t) = R sin(2t) + yc ,

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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

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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)

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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

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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

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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|

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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

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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).

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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))

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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)))

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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

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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)

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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 = ?

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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

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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 ?

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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.

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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.

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Representing complex curves

  • Typically represented

    • A series of simpler curve (each defined by a single equation) pieced together at their endpoints.(piecewise construction)

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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).

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An interactive curve design

a) Desired curve

b) User places points

c) The algorithm generates many points along a “nearby” curve

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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.

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