Lecture 9 Notes:
Blending Surfaces
9.1
Examples and motivation
Blending surfaces, providing a smooth connection between various primary or functional surfaces, are very common in CAD. Examples include blending surfaces between:
_ Fuselage and wings of air
Lecture 3 Notes:
Dierential geometry of surfaces
3.1
De_nition of surfaces
_ Implicit surfaces F (x; y; z) = 0
Example: a2 + b2 + c2 = 1 Ellipsoid, see Figure 3.1.
y
z
x
Figure 3.1: Ellipsoid.
_ Explicit surfaces
If the implicit equation F (x; y; z) = 0 c
Lecture 8 Notes:
Fitting, Fairing and Generalized
Cylinders
8.1
Least Squares Method of Curve Fitting
Example problem
Given N points Pi , i = 1, 2, ., N (N 4), construct an approximating cubic B
ezier curve that interpolates P1 and PN (end points).
Soluti
Lecture 6 Notes:
B-splines (Uniform and Non-uniform)
6.1
Introduction
The formulation of uniform B-splines can be generalized to accomplish certain objectives.
These include
_ Non-uniform parameterization.
_ Greater general
exibility.
_ Change of one poly
Lecture 4 Notes:
Introduction to Spline Curves
4.1
Introduction to parametric spline curves
Parametric formulation
x = x(u); y = y(u); z = z(u)
or
R = R(u)
(vector notation)
Usually applications need a _nite range for u (e.g. 0 _ u _ 1).
For free-form sha
Lecture 1 Notes:
Introduction and classication of
geometric modeling forms
1.1
Motivation
Geometric modeling deals with the mathematical representation of curves, surfaces, and solids
necessary in the denition of complex physical or engineering objects. T
Lecture 19 Notes: Decomposition
models
19.1
Introduction
Decomposition models are representations of solids via combinations (unions) of basic special
building blocks glued together. Alternatively, decomposition models may be considered to
represent solid
Lecture 20 Notes:
Advanced topics in dierential
geometry
20.1
Geodesics
In this section we study the computation of shortest path between two points on free-form
surfaces [14, 11].
20.1.1
Motivation
ship design
robot motion planning
terrain navigation
Lecture 13 Notes:
Osets of Parametric Curves and
Surfaces
13.1
Motivation
Osets are de_ned as the locus of points at a signed distance d along the normal of a planar
curve or surface. A literature survey on oset curves and surfaces up to 1992 was carried
Lecture 2 Notes:
Dierential geometry of curves
2.1
De_nition of curves
2.1.1
Plane curves
_ Implicit curves f (x; y) = 0
Example: x2 + y 2 = a2
cfw_ It is di_cult to trace implicit curves.
cfw_ It is easy to check if a point lies on the curve.
cfw_ Multi-