1. Basic theory of curve and surface

2. Geometric representation
Parametric
Non-parametric
Explicit
Implicit
x = x(u), y = y(u)
y = f(x)
f(x, y) = 0

3. Geometric representation
Example - circle
Parametric
Non-parametric
Explicit
Implicit
x = R cos, y = R sin
y = R2 – x2
x2 + y2 – R2 = 0

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

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

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

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

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

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.

10. Parametric (cont) Advantages (cont)
easy to express in the form of vectors and matrices
Inherently bounded.
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11. 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 t3 t2 t4 t5 t6 t1 x

12. Parametric curve
Fundamental geometric objects – lines, rays and line segment a b
line a b
ray a b
Line segment
All share the same parametric representation

13. 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   a b

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 = t

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

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

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

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.
ellipse
hyperbola
parabola

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

20. Parametric curve (circle)
If t =  a 1/8 circle
Circular arc
t =  a 1/4 circle
t = 0.5  a ½ circle
t = 1  a circle

21. Parametric curve (circle)
Circle with center at (xc, yc)
x(t) = R cos(2t) + xc,
y(t) = R sin(2t) + yc ,

22. Parametric curve Ellipse Hyperbola parabola b 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

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

25. Tangent vector and tangent line
Vector tangent to the slope of curve at a given point
Tangent line
The line that contains the tangent vector

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|

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

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).

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))

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)))

31. Example Check / prove Let say, t = 0, R
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

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)

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 = ?

34. 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

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 ?

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.

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.

38. Representing complex curves
Typically represented
A series of simpler curve (each defined by a single equation) pieced together at their endpoints.(piecewise construction)

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).

40. An interactive curve design
a) Desired curve
b) User places points
c) The algorithm generates many points along a “nearby” curve

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.