This document discusses different methods for representing curves in geometric modeling, including non-parametric and parametric representations. Non-parametric representations can be explicit, using equations relating x, y, and z, or implicit, with no distinction between dependent and independent variables. Parametric representations describe variables in terms of a parameter, allowing for multi-valued and flexible curves. Specific curve types discussed include Bezier curves, defined by control points, and B-spline curves, which separate the polynomial order from the number of control points.

1. Geometric Modeling
Unit No. 02

2. Introduction
• Can be described by arrays of coordinate data
or by analytical data
• It is important entities in geometric modeling
Curves Entities
Analytical Entities Synthetic Entities

3. Representation of Curves
Representation of Curves
Non Parametric Parametric

4. Non Parametric Representation of Curve
• The curve is represented as a relationship
between x, y and z
• There are two types of non parametric
representation
1. Explicit Non Parametric Representation
2. Implicit Non Parametric Representation

5. Conti….
• Explicit non-Parametric Equation
y = c1 + c2 x + c3 x2 + c4 x3
There is a unique single value of the dependent
variable for each value of the independent
variable.
• Implicit non-parametric equation
(x – xc)2 + (y – yc)2 = r2
No distinction is made between the dependent and
the independent variables.

6. Parametric Equation
• Describe the dependent and independent variables
in terms of a parameter
• Can be converted to a non-parametric form, by
eliminating the dependent and independent
variables from the equation
• Allow great versatility in constructing space
curves that are multi-valued and easily
manipulated
• Parametric curves can be defined in a constrained
period (0 ≤ t ≤ 1)
x = r cosθ, y = r sinθ

7. Bezier Curve
• Difficult to change the shape of Hermite Cubic
Spline
• Defined by set of Data Points
• Curve may interpolate or Extrapolate the data
points

8. Characteristics of Bezier Curve
• Shape of the Bezier curve is controlled by its
defining points. Tangent vectors are not used in
the development of curve as in case of Cubic
spline
• The order or Degree of Bezier curve is variable
and is related to number of points defining it.
• (n+1)th points define nth degree curve, which
permits higher order continuity.
• Data points of Bezier curve are called as Control
Points

9. Bezier Curve

10. Conti……

11. Cubic Bezier Curve for various Control Points

12. B Spline Curves
• Another Method to generating a curve defined by
data points
• It is proper and powerful generalization of Bezier
Curve
• Problem associate with Bezier curve is, with as
increase in number of data points, the order of
polynomial representing the curve is increases.
• B Spline curve separates the order of polynomial
representing the curve from number of given data
points

13. Advantage
• B spline curve allows local control over the shape
of curve as against the global control in Bezier
curve
• Degree of polynomial representing the curve can
be set independently of number of control points
• B Spline curves gives better control
• It permits add or delete any number of control of
data point without changing the degree of
polynomial