Curves

1. CURVES -
REPRESENTATION

2. Many technological applications
 Design of products (e.g. CAD)
 Calculation of the path for a robot
Design of fonts
 Large sized fonts must be smooth
Interpolating measuring data
Approximating measuring data
Why designing curves?

3. Controllability
 Changes must be predictable in effect
 Intuitive to use for the designer
Locality
 Local changes should stay local
Smoothness
 No sharp bends
Criterias for curves

4.  3 basic representation strategies:
 Explicit: y = mx + b
 Implicit: ax + by + c = 0
 Parametric: P = P0 + t (P1 - P0)
Curve Representations

5.  More degrees of freedom
 Directly transformable
 Dimension independent
 No infinite slope problems
 Separates dependent and independent variables
 Inherently bounded
 Easy to express in vector and matrix form
 Common form for many curves and surfaces
Advantages of parametric forms

6.  Spline curve
 Convex hull
 Control graph
 Piecewise cubic splines
Spline Representations

7.  Smooth curve that is defined by a sequence of points.
Spline curve
Interpolating spline Approximating spline

8.  Smallest polygon that encloses all points
Convex hull
Interpolating spline Approximating spline
Convex hull

9.  Polyline through sequence of points
Control graph
Interpolating spline Approximating spline
Control graph

10. Piecewise cubic splines
Segments

11. Interpolation
 Very bad locality
 Tend to oscillate
 Small changes may result in
catastrophe
 Bad controllability
 All you know is, that it
interpolates the points
 High effort to evaluate curve
 Imagine a curve with several
million given points
Interpolation vs Approximation
Approximation
 Unlike interpolation the points are
not necessarily interpolated
 Points give a means for
controlling of where the curve
goes
 Often used when creating the
design of new (i.e. non-existing)
things
 No strict shape is given

12.  Parametric continuity Cx
 Only P is continuous: C0
 Positional continuity
 P and first derivative dP/du are continuous: C1
 Tangential continuity
 P + first + second: C2
 Curvature continuity
 Geometric continuity Gx
 Only directions have to match
Continuity in Curves - Representation

13. Parametric continuity Cx - Order of continuity
Zero-order parametric continuity
C0: P(1) = Q(0).
Endpoint of P(u) coincides with start point Q(v).
P(u) Q(v)
First order parametric continuity
C1: dP(1)/du = dQ(0)/dv.
Direction of P(1) coincides with direction of Q(0).
First order parametric continuity gives a smooth
curve. Sometimes good enough, sometimes not.
P(u) Q(v)

14. Contd..
Second order parametric continuity
C2: d2P(1)/du2 = d2Q(0)/dv2.
Curvatures in P(1) and Q(0) are equal.
P(u) Q(v)

15. Geometric continuity Gx
 Here the vectors are exactly equal.
 It suffices to require that the directions are the same.
First order geometric continuity:
G1: dP(1)/du =  dQ(0)/dv with  >0.
Direction of P(1) coincides with direction Q(0).
P(u) Q(v)

16. THANK YOU