1. Surfaces and Curves

2. Madhulika (18010), Assistant Professor, LPU.
Introduction
Graphics scenes can contain many different kinds of objects
No one method can describe all these objects
Accurate models produce realistic displays of scenes
Polygon and quadric surfaces
Spline surfaces and construction techniques
Procedural methods
Physically-based modeling methods
Octree encodings
Boundary representations (B-reps)
Space-partitioning representations
Madhulika (18010), Assistant Professor, LPU.

3. Madhulika (18010), Assistant Professor, LPU.
Polygon Surfaces
The most commonly used boundary represent-ation
All surfaces are described with linear equations
Simplify and speed up surface rendering and display of objects
Precisely define a polyhedron
For non-polyhedron objects, surfaces are tesselated
Commonly used in design and solid modeling
The wireframe outline can be display quickly
Madhulika (18010), Assistant Professor, LPU.

4. Madhulika (18010), Assistant Professor, LPU.
Polygon Tables
Specify a polygon surface
Geometric tables
Vertex coordinates
Spatial orientation of the polygon
Attribute tables
Degree of transparency
Surface reflectivity
Texture
Madhulika (18010), Assistant Professor, LPU.

5. Madhulika (18010), Assistant Professor, LPU.
Polygon Tables
Convenient organization for geometric data
Vertex Table
Edge Table
Polygon Table v1 E3 E6 E1 S1 S2 v3 v5 E2 E4 E5 v2 v4
Madhulika (18010), Assistant Professor, LPU.

6. Madhulika (18010), Assistant Professor, LPU.
Polygon Tables
Madhulika (18010), Assistant Professor, LPU.

7. Madhulika (18010), Assistant Professor, LPU.
Polygon Tables
Alternative arrangements
Use just two tables: a vertex table and a polygon table
Some edges could get drawn twice
Use only a polygon table
This duplicates coordinate information
Extra information to the data tables
Expand the edge table to include forward pointers into the polygon table
Common edges between polygons
Expand vertex table to contain cross-reference to corresponding edges
Madhulika (18010), Assistant Professor, LPU.

8. Madhulika (18010), Assistant Professor, LPU.
Polygon Tables
Additional geometric information
The slope for each edge
Coordinate extents for each polygon
Tests for consistency and completeness
Every vertex is an endpoint for at least two edges
Every edge is part of at least one polygon
Every polygon is closed
Each polygon has at least one shared edge
Every edge referenced by a polygon pointer has a reciprocal pointer back to the polygon
Madhulika (18010), Assistant Professor, LPU.

9. Madhulika (18010), Assistant Professor, LPU.
Plane Equations
The equation for a plane surface
Ax + By + Cz +D = 0
By Cramer’s rule:
Values of A,B,C and D are computed for each polygon and stored with the other polygon data
Madhulika (18010), Assistant Professor, LPU.

10. Madhulika (18010), Assistant Professor, LPU.
Plane Equations
Orientation of a plane surface
Surface normal vector (A, B, C)
Distinguish two sides of a surface
“Inside” face
“Outside” face
The normal vector from inside to outside
Viewing the outer side of the plane in a right-handed coordinate system
Vertices are specified in counterclockwise direction
Madhulika (18010), Assistant Professor, LPU.

11. Madhulika (18010), Assistant Professor, LPU.
Plane Equations
N=(A,B,C)
Madhulika (18010), Assistant Professor, LPU.

12. Madhulika (18010), Assistant Professor, LPU.
Plane Equations
Madhulika (18010), Assistant Professor, LPU.

13. Madhulika (18010), Assistant Professor, LPU.
Plane Equations
Identify a point (x, y, z) as either inside or outside a plane surface
In a right-handed coordinate system, the plane parameters A, B, C, and D were calculated using vertices selected in a counterclockwise order when viewing the surface in an outside-to-inside direction.
If Ax+By+Cz+D<0, the point(x,y,z) is inside the surface
If Ax+By+Cz+D>0, the point(x,y,z) is outside the surface
Madhulika (18010), Assistant Professor, LPU.

14. Madhulika (18010), Assistant Professor, LPU.
Curves
Although used for display, bit maps are not appropriate for most computational tasks, which need a three-dimensional representation of the objects composing the image.
Represented skeletally as a wire-frame image, the Utah Teapot is composed of many small polygons. However, even with hundreds of polygons, the image is not smooth.
Smoother representations can be provided by Bezier curves, which have the further advantage of requiring less computer memory.
Madhulika (18010), Assistant Professor, LPU.

15. Madhulika (18010), Assistant Professor, LPU.
Bezier curves are described by cubic equations; a cubic curve is determined by four points or, equivalently, by two points and the curve’s slopes at those points.
Two cubic curves can be smoothly joined by giving them the same slope at the junction.
Bezier curves, and related curves known as B-splines, were introduced in computer-aided design programs for the modeling of automobile bodies.
Madhulika (18010), Assistant Professor, LPU.

16. Madhulika (18010), Assistant Professor, LPU.
Spline
To produce a smooth curve through a designed set of points, a flexible strip called spline is used.
Such a spline curve can be mathematically described with a piecewise cubic polynomial function .
Madhulika (18010), Assistant Professor, LPU.

17. Madhulika (18010), Assistant Professor, LPU.
Big Idea
User specifies control points
Defines a smooth curve
Curve
Control Points
Control Points
Madhulika (18010), Assistant Professor, LPU.

18. Interpolation Vs Approximation
A spline curve is specified using a set of control points
There are two ways to fit a curve to these points:
Interpolation - the curve passes through all of the control points
Approximation - the curve does not pass through all of the control points
Madhulika (18010), Assistant Professor, LPU.

19. Madhulika (18010), Assistant Professor, LPU.
Bézier Spline Curves
A spline approximation method developed by the French engineer Pierre Bézier (1962)for use in the design of Renault car bodies
A Bézier curve can be fitted to any number of control points – although usually 4 are used
Madhulika (18010), Assistant Professor, LPU.

20. Madhulika (18010), Assistant Professor, LPU.
Bezier splines :useful and convenient for curve and surface design
A Bezier curve is a parametric curve frequently used in computer graphics and related fields.
Generalizations of Bezier curves to higher dimensions are called Bezier surfaces, of which the Bezier triangle is a special case.
Bezier curves are used to model smooth curves that can be scaled indefinitely.
Madhulika (18010), Assistant Professor, LPU.

21. Bézier Spline Curves (cont…)
Consider the case of n+1 control points denoted as pk=(xk, yk, zk) where k varies from 0 to n
The coordinate positions are blended to produce the position vector P(u) which describes the path of the Bézier polynomial function between p0 and pn
Madhulika (18010), Assistant Professor, LPU.

22. Bézier Spline Curves (cont…)
The Bézier blending functions BEZk,n(u) are the Bernstein polynomials
where parameters C(n,k) are the binomial coefficients
Madhulika (18010), Assistant Professor, LPU.

23. Bézier Spline Curves (cont…)
These polynomials give the blending polynomials for any degree Bezier form:
All zeros at 0 and 1.
For any degree they all sum to 1.
They are all between 0 and 1 inside (0,1) .
Madhulika (18010), Assistant Professor, LPU.

24. Bézier Spline Curves (cont…)
So, 3 parametric equations for the individual curve coordinates
Madhulika (18010), Assistant Professor, LPU.

25. Madhulika (18010), Assistant Professor, LPU.
Bezier curve is polynomial of degree 1 less than number of control points used. 3 points generate parabola,4 points give cubic curve and so on
2d Bezier curves generated from 3,4 and 5 control points.
Dashed lines connect control point positions
Madhulika (18010), Assistant Professor, LPU.

26. Important Properties Of Bézier Curves
The curve passes through the first, P0 and last vertex points, Pn
or first and last control points are the first and last point on the curve
P(0) = p0
P(1) = pn
The points Pi are called control points for the Bézier curve.
The polygon formed by connecting the Bézier points with lines, starting with P0 and finishing with Pn, is called the Bézier polygon.
The convex hull(convex polygon boundary) of the Bézier polygon contains the Bézier curve.The curve lies within the convex hull of control points. Bézier blending functions are all positive and sum to 1
Means curve position is weighted sum of control points positions. It ensures that polynomial smoothly follows control points without oscillations.
Madhulika (18010), Assistant Professor, LPU.

27. Madhulika (18010), Assistant Professor, LPU.
Convex Hull Property
The properties of the Bernstein polynomials ensure that all Bezier curves lie in the convex hull of their control points.
Hence, even though we do not interpolate all the data, we cannot be too far away. p1 p2
convex hull
Bezier curve p3 p0
Madhulika (18010), Assistant Professor, LPU.

28. Madhulika (18010), Assistant Professor, LPU.
properties
3. The tangent vector at the starting point P0 must be given by P1 – P0 and the tangent Pn given by Pn – Pn-1
Values of the parametric first derivatives of bezier curve at end points can be calculated from control point coordinates as:
The slope at the beginning of the curve is along the line joining the first two control points.
The slope at the end of the curve is along the line joining the last two control points
4. Similarly, parametric second derivatives of bezier curve at end points can be calculated.This requirement is generalized for higher derivatives at the curve’s end points. E.g 2nd derivative at P0 can be determined by P0 ,P1 ,P2 (to satisfy continuity)
5. The same curve is generated when the order of the control points is reversed
Madhulika (18010), Assistant Professor, LPU.

29. Madhulika (18010), Assistant Professor, LPU.
Bezier Curves
The degree of the Bezier polynomial
One less than the number of control points used.
The relative position of control points
With certain placements, we obtain degenerate Bezier polynomials.
Madhulika (18010), Assistant Professor, LPU.

30. Design Techniques Using Bezier Curves
Closed Bezier curves
Specifying the first and last control points at the same position.
Multiple control points
Specifying multiple control points at a single coordinate position gives more weight to that position.
Madhulika (18010), Assistant Professor, LPU.

31. Madhulika (18010), Assistant Professor, LPU.
Cubic Bezier Curves
Four blending functions for cubic Bezier curves
Blending function BEZ 1,3 is maximum at u = 1/3.
Blending function BEZ 2,3 is maximum at u = 2/3.
Bezier curves do not allow for local control of the curve shape.
Madhulika (18010), Assistant Professor, LPU.

32. Madhulika (18010), Assistant Professor, LPU.
Cubic Bezier Curves
Madhulika (18010), Assistant Professor, LPU.

33. Madhulika (18010), Assistant Professor, LPU.

34. Madhulika (18010), Assistant Professor, LPU.
Cubic Bezier Curves
Madhulika (18010), Assistant Professor, LPU.

35. Madhulika (18010), Assistant Professor, LPU.
Bezier Surfaces
The Bezier surface
Its parametric vector function is formed as the Cartesian product of Bezier blending functions
Madhulika (18010), Assistant Professor, LPU.

36. Madhulika (18010), Assistant Professor, LPU.
The form of blending function determine how the control points influence shape of curve for values of parameter u over range from 0 to 1.
At u=0, blending function is BEZ 0,3 ,which has value 1.
At u=1, blending function is BEZ 3,3 ,which has value of 1 at that point.
So, Cubic bezier curve will always pass through control points p0 and p3.
Other fuctions BEZ 1,3 and BEZ 2,3 influence shape of curve at intermediate values of parameter u,so curve tends towards points p1 and p2.
Blending function BEZ 1,3 is maximum at u = 1/3.
Blending function BEZ 2,3 is maximum at u = 2/3.
Madhulika (18010), Assistant Professor, LPU.

37. Madhulika (18010), Assistant Professor, LPU.
Disadvantages
The degree of the Bezier curve depends on the number of control points.
The Bezier curve lacks local control. Changing the position of one control point affects the entire curve.
Madhulika (18010), Assistant Professor, LPU.

38. Madhulika (18010), Assistant Professor, LPU.
Summary
Today we had a look at spline curves and in particular Bézier curves
The whole point is that the spline functions give us an approximation to a smooth curve
Madhulika (18010), Assistant Professor, LPU.

39. Madhulika (18010), Assistant Professor, LPU.
Problem
Find equation of bezier curve which passes through points (0,0) and (-2,1) and is controlled through points (7,5) and (2,0).
Madhulika (18010), Assistant Professor, LPU.

40. Madhulika (18010), Assistant Professor, LPU.

41. Madhulika (18010), Assistant Professor, LPU.
B-Spline Curves
B-splines have two advantages over Bezier splines:
The degree is independent of the number of control points.
Allow local control of the shape.
Blending function formulation
The range of parameter u depends on how we choose the B-spline parameters.
B k,d are polynomials of degree d - 1.
Madhulika (18010), Assistant Professor, LPU.

42. Madhulika (18010), Assistant Professor, LPU.
B-Spline Curves
B-spline blending functions
Cox-deBoor recursion formulas:
Knot vector: The selected set of subinterval endpoints u j.
We can choose any value for the subinterval endpoints satisfying the relation u j+1  u j
Madhulika (18010), Assistant Professor, LPU.

43. Madhulika (18010), Assistant Professor, LPU.
B-Spline Curves
Madhulika (18010), Assistant Professor, LPU.

44. Madhulika (18010), Assistant Professor, LPU.
B-Spline Curves
Values for umin and umax depend on:
The number of control points we select.
The value we choose for parameter d.
How we set up the subintervals.
Madhulika (18010), Assistant Professor, LPU.

45. Madhulika (18010), Assistant Professor, LPU.
B-Spline Curves
Properties of B-spline curves
Have degree d - 1 and C d-2 continuity over the range of u.
For n + 1 control points, the curve is described with n + 1 blending functions.
Each B k,d is defined over d subintervals, starting at knot value u k.
The range of parameter u is divided into n + d subintervals by n + d + 1 knot values.
With knot values {uo, u1, ..., u n+d}, the resulting B-spline curve is defined from u d-1 to u n+1.
Any one control point can affect the shape of at most d curve sections.
Madhulika (18010), Assistant Professor, LPU.

46. Madhulika (18010), Assistant Professor, LPU.
B-Spline Curves
Convex Hull property
The B-spline curve holds the property.
For any parameter value of u in the interval from knot value u d-1 to u n+1, the sum over all basis functions is 1.
For different knot vectors, there are three classes:
Uniform
Open uniform
Non-uniform
Madhulika (18010), Assistant Professor, LPU.

47. Uniform, Periodic B-Splines
Uniform B-spline
The spacing between knot values is constant.
{-1.5,-1.0,-0.5,0.0,0.5,1.0,1.5,2.0}
Often normalized to the range between 0 and 1.
{0.0,0.2,0.4,0.6,0.8,1.0}
Uniform B-splines have periodic blending functions.
Periodic blending function
For given values of n and d, all blending functions have the same shape.
Each successive blending function is simply a shifted version of the previous function:
Madhulika (18010), Assistant Professor, LPU.

48. Uniform, Periodic B-Splines
Madhulika (18010), Assistant Professor, LPU.

49. Uniform, Periodic B-Splines
Madhulika (18010), Assistant Professor, LPU.

50. Uniform, Periodic B-Splines
Example: d = n = 3
Knot vector = {0, 1, 2, 3, 4, 5, 6}
The slope at the start position is parallel to the line joining the first two control points.
The slope at the end position is parallel to the line joining the last two control points. p1 p2
Pstart =1/2(P0+P1)
Pend =1/2(P2+P3)
P’start = P1 -P0 p0
P’end = P3 -P2 p3
Madhulika (18010), Assistant Professor, LPU.

51. Cubic, Periodic B-Splines
For cubics, d = 4 and each blending function spans four subintervals. If we are to fit the cubic control points,then we could use the integer vector
{0,1,2,3,4,5,6,7}
An alternate formulation:
start with the boundary conditions,
normalized interval u[0,1]
Madhulika (18010), Assistant Professor, LPU.

52. Cubic, Periodic B-Splines
A matrix formulation for a cubic periodic B-spline with four control points
Madhulika (18010), Assistant Professor, LPU.

53. Cubic, Periodic B-Splines
Example:
A closed, periodic, piecewise, cubic B-spline constructed with cyclic specification of the six control points.
Madhulika (18010), Assistant Professor, LPU.

54. Open, Uniform B-Splines
A class between uniform and nonuniform B-splines
Knot spacing is uniform except at the ends where knot values are repeated d times.
Example:
{0, 0, 1, 2, 3, 3}, for d = 2 and n = 3
{0, 0, 0, 0, 1, 2, 2, 2, 2}, for d = 4 and n = 4
For any d and n, the knot vector
Madhulika (18010), Assistant Professor, LPU.

55. Open, Uniform B-Splines
Characteristics are very similar to Bezier splines
When d = n + 1, open B-splines reduce to Bezier splines, all knot values are either 0, or 1.
Example: a cubic, open B-splines (d = 4)
Knot vector = {0, 0, 0, 0, 1, 1, 1, 1}
An open B-spline passes through the first and last control points.
Specifying multiple control points at the same coordinate position pulls curve closer.
Madhulika (18010), Assistant Professor, LPU.

56. Madhulika (18010), Assistant Professor, LPU.
Nonuniform B-Splines
Nonuniform B-splines
We can specify any values and intervals for the knot vector.
{0, 1, 2, 3, 3, 4}
{0, 2, 2, 3, 3, 6}
{0, 0.2, 0.6, 0.9, 1.0}
Provides increased flexibility in controlling a curve shape.
Multiple knot values reduce the continuity by 1 for each repeat of a particular value.
Madhulika (18010), Assistant Professor, LPU.

57. Madhulika (18010), Assistant Professor, LPU.
B-Spline Surfaces
The Cartesian product of B-spline blending functions in the form:
Where are (n1 + 1) by (n2 + 1) control points
Madhulika (18010), Assistant Professor, LPU.

58. Madhulika (18010), Assistant Professor, LPU.
Bezier vs. B-Spline
Madhulika (18010), Assistant Professor, LPU.

59. Madhulika (18010), Assistant Professor, LPU.