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Computer Graphics
Week 10 Lecture 1

2. Splines
(Modelling Curves)

3. Question
Before the advent of sophisticated Computers how did people draw curves accurately ?

4. Spline : Flexible strip used to draw curves through designated points

5. Splines
Originally , the curves drawn with spline were called Spline curves.
Splines are now used to model a variety of curves (natural and artificial)
We usually mathematically model them them using piecewise cubic functions
spline curve now refers to any composite curve formed with polynomial sections satisfying any specified continuity conditions at the boundary of the pieces
(1st and 2nd derivatives are continuous)

7. Uses of Splines
Use of approximate naturally occurring curves (landscape, leaves, astronomical bodies , electron orbitals)
Design curves
Digitize drawings
Specify animation paths
CAD uses splines for design of automobile bodies , surfaces of aircraft and space crafts, ship hulls and home appliances

8. We specify Splines by a set of coordinate points called Control Points , which indicate the general shape of the curve.
Based on how control points are used , there are 2 drawing techniques:
1.Interpolation
2.Approximation

9. Interpolation and Approximation
When polynomial sections are fitted such that all control points are connected
When a polynomial curve is fitted such that some or all the control points are not on the generated curve
The control points specify the general path of the curve

10. Parametric continuity Constraints
To ensure smooth transition from one section of spline to the next we impose various continuity constraints
A cubic curve section can be parameterized as follow:

11. Zero Order Parametric Continuity
represented as C0 , it simply means that the two curves meet at a common point.
The endpoint of first curve is the starting point of the successive curve
First Order Parametric Continuity
represented as C1 ,
Here Additionally , the first parametric derivative (tangent) of the two curves is same at the meeting point
Second Order Parametric Continuity
represented as C2 ,
Here Additionally , the second parametric derivative of the two curves is same at the meeting point as well
i.e. the rate of change of tangent of the 2 curves is also same

12. There are 3 equivalent methods for specifying a spline:
Stating set of boundary conditions on the spline
Stating a matrix that specifies a spline
Stating set of blending functions (or basis functions) that determine how specified constraints are combined to calculate curve
Spline
Specification

13. Cubic interpolation methods
In these methods the piecewise curves are cubic polynomials
Benefits of cubic polynomials → Speed and flexibility
Given n+1 control points :
Our need is to fit cubic curves between adjacent control points. For each curve we can define coordinates that lie on the curve in cubic parametric eq:

14. Contd...
We can find equation of each curve by finding the values of coefficients a,b,c,d for each piecewise curve
We impose continuity contraints to find these coefficients

15. Interpolation Methods
We will look at 2 interpolation methods:
Natural Cubic Interpolation (the simplest one)
Hermite Interpolation

16. Natural Cubic Interpolation
Simplest method
Suppose we have n+1 control points ⇒ We need to draw n curves
For each curve we need to know 4 coefficients (a,b,c,d) → Need 4 equations (or 4 boundary conditions)
Conditions on board
No “local” control

17. Hermite Method Developed by French Mathematician, Charles Hermite
Allow for more local control
Require specified tangent at each control point

18. Hermite Method Here P(u) → eq. of curve section , pk → control point k
Then in Hermite method the boundary conditions that define the curve are:

19. Hermite Method(matrix form)
The curve equation :
can be expressed in vector form as :
which can be written in matrix form as (eq.1):

20. contd... Taking derivative wrt u:
Now since we know P(u) and P’(u) we can rewrite our boundary conditions as:

21. contd...
Putting coefficients value back in eq1 :

22. Hermite Method (blending func. form)
Multiplying the previous matrix equation we get:
Here H0,H1,H2,H3 are Hermite blending functions.

23. contd...

24. Hermite Method Issue with Hermite function Variation :
Cardinal splines
Kochanek-Bartels

25. The End

26. Splines
Originally , the curves drawn with spline were called Spline curves.
Splines are now used to model a variety of curves (natural and artificial)
We mathematically model them them using

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–Johnny Appleseed