1. Three Dimensional Object Representation

2. Modeling of object or picture means describing
them to the computer so as to produce visual
display that simulate the real thing
For describing we need a set of
primitives /geometric forms
Complex
Simple
Curved Segment
Curved Surface
Quadratic Surface
Points
Lines
Poly lines
Polygon
Polyhedron
Highly realistic representation tse color, shading & texture

3. Point
Pt p in 2D p(x,y) and in 3D p(x,y,z)
Line
In 2D it specified by end pt (x1,y1) to (x2,y2)
In 3D it specified by end pt (x1,y1,z1) to (x2,y2,z2)
Poly Line
Chain of connected line segments from pt p0 to pn
Polygons
A closed poly lines where initial pt & final pt coincide. It is specified by vertex & edge information
Planner polygon – When all vertex lie on same plane
Polyhedron
A closed polygonal net in which each polygon (face) is planner.
It is solid object

4. Wireframe Model Representation of polygonal net/ polygonal mesh
Here representation of any 3D object can be done by vertex, edge and polygon information
In 3 ways we can represent
Explicit vertex list
Polygon Listing
Explicit edge Listing

5. Representation of this objectD
Problem
Representation of this object H E B A
Explicit vertex list G F
Here the information of the object will be stored in the order in which they will be encountered by travelling around the complete model
Let the travelling sequence is ---
A -> B -> C -> D ->A -> F -> E -> D -> C
->H ->E -> F -> G -> H->G ->B C D H E B A G F

6. Disadvantage
Inefficient for a complete polygonal net where shared
vertices are repeated several time.
( More space complexity)
For displaying the polygonal net from this type of
stored information shared edges are drawn several times
(More time complexity)
Advantage
It is useful for single polygon.

7. Here each vertex is stored exactly once in vertex list V= (v0 ..vn)
Polygon listing
Here each vertex is stored exactly once in vertex list
V= (v0 ..vn)
Here each polygon is stored exactly once in polygon list
by pointing or indexing to the vertex list
The vertex list is V={A,B,C,D,E,F,G,H}
The polygon list is -- P= {p1,p2,p3,p4,p5,p6}
P1 ={A,B,C,D}  {V(0),V(1),V(2),V(4)}
P2 = {C,D,E,H} 
P3 = {A,B,E,F} 
P4 = {A,B,G,F} 
P6 ={E,F,G,H}  {V(4),V(5),V(6),V(7)} D C H E B A G F

8. Disadvantage
Shared edges are drawn several times in displaying
through this method
(time complexity)
Advantage
More efficient than explicit vertex list

9. Here each vertex is stored exactly once in vertex list V= (p0 ..pn)
Edge listing
Here each vertex is stored exactly once in vertex list
V= (p0 ..pn)
Each edge is stored exactly once in edge list by pointing to the 2 vertex of vertex list for defining every edge.
Each polygon is stored exactly once in polygon list
by pointing or indexing to the edge list
Drawing will be done by considering the edge list C D H E B A G F

10. Advantage (of wire frame model)
Disadvantage
none
Advantage
used to represent more general wireframe model
Advantage (of wire frame model)
Wire frame model are used in engineering application
As they are composed of straight lines
 Easy to construct
 Easy to clip & manipulate
Disadvantage (of wire frame model)
For achieving the realistic model which are highly
curved object , here huge no of polygons are needed

11. Plane Equation
To produce a display of 3 dimensional object, we need the information of spatial orientation of the individual surface components of the object. This information is obtained from vertex coordinate value and equation that describes the polygon plane. The equation of plane surface:- where (x, y, z) is any point on the plane. and A,B,C,D describes spatial property of the plane, (whose value can be obtained by solving a set of 3 plane equations using the coordinate values for 3 nonlinear points in the plane).
Ax + By + Cz + D =0

12. the point (x,y,z) is inside the surface
We need to distinguish the side of the polygon surfaces which enclose the object interior
Inside face:- The side of the plane that faces the object interior.
Outside face :- The side of the plane that faces the object exterior or visible.
The position of a spatial point (x,y,z) relative to plane surface of an object can be identified using plane equation.
the point (x,y,z) is inside the surface
the point (x,y,z) is outside the surface
Ax + By + Cz + D <0
Ax + By + Cz + D >0

13. Why have curves ? Representation of “irregular surfaces” Example:
Artist’s representation
Auto industry (car body design)
Clay / wood models
Tool and die Manufacturing

14. “Computers can’t draw curves.”
The more points/line segments that are used, the smoother the curve.

15. Classification of Curves
Analytical Curves: This type of curve can be represented by a simple mathematical equation, such as, a circle or an ellipse. They have a fixed form and cannot be modified to achieve a shape that violates the mathematical equations
Interpolated curves: An interpolated curve is drawn by interpolating the given data points and has a fixed form, dictated by the given data points. These curves have some limited flexibility in shape creation, dictated by the data points.
Approximated Curves: These curves provide the most flexibility in drawing curves of very complex shapes. The model of a curved automobile fender (a plastic cylinder, tyre)can be easily created with the help of approximated curves and surfaces.

16. Parametric and Non-parametric Equations of a Curve
The mathematical representation of a curve can be classified as either parametric or nonparametric
1 Nonparametric: -
Explicit non-parametric eqtation :
y = mx + c
y = c1 + c2 x + c3 x 2 + c4 x 3
There is a unique single value of the dependent variable for each value of the independent variable.
Implicit non-parametric equation :
Ax + By + C = 0
(x – xc)2 + (y – yc)2 = r2
No distinction is made between the dependent and the independent variables

17. 2. Parametric: Parametric equations describe the dependent and independent variables in terms of a parameter. Parametric curves can be defined in a constrained period (0 ≤ t ≤ 1); since curves are usually bounded in computer graphics, this characteristic is of considerable importance Line :- x = x1 + (x2 - x1) t where, 0 ≤ t ≤ 1 y = y1 + (y2 - y1) t circle :- x = r cosθ, y = r sinθ where 0 ≤ θ ≤ 2π Ellipse:- x = rx cosθ, y = ry sinθ

18. Curve representation
Problem: How to represent a curve easily and efficiently
“Brute force and ignorance” approaches:
storing a curve as many small straight line segments
doesn’t work well when scaled
inconvenient to have to specify so many points
need lots of points to make the curve look smooth
working out the equation that represents the curve
difficult for complex curves
moving an individual point requires re-calculation of the entire curve

19. Requirements Want mathematical smoothness Local control
Some number of continuous derivatives of P
Local control
Local data changes have local effect
Continuous with respect to the data
No wiggling if data changes slightly
Low computational effort

20. A solution use SEVERAL polynomials
Complete curve consists of several pieces
All pieces are of low order
Third order is the most common
Pieces join smoothly
This is the idea of spline curves
or just “splines”

23. Parametric cubic curves
General form:

24. General curve shape may be generated using method of
Interpolation (also known as curve fitting)
Curve will pass through control points
Approximation
Curve will pass near control points may interpolate the start and end points.

25. Curve algorithm
Interpolation
Approximation

26. Interpolation vs approximation
f(x)
curve most pass through control points
curve is influenced by control points x
f(x)

27. Interpolation Curves
Curve is constrained to pass through all control points
Given points P0, P1, ... Pn, find lowest degree polynomial which passes through the points x(t) = an-1tn a2t2 + a1t + a0 y(t) = bn-1tn b2t2 + b1t + b0

28. Interpolating curve : piecewise linear
Example: Hermite Curve
Curve defined by multiple segments (linear) which are defined by polynomial functions (cubic functions).
Segments joints known as KNOTS
Requires too many data points for most shape
Representation not flexible enough to editing

29. Approximation techniques
Developed to permit greater design flexibility in the generation of free form curves
Common methods in modern CAD systems, bezier, b-spline, NtRBS
Employ control points (set of vertices that approximate the curve)

30. Approximation techniques
Curves do not pass directly through points (except start and end)
Intermediate points affect shape as if exerting a “pull” on the curve.
Allow user to set shape by “pulling” out curve using control point location.

31. Example – bezier curve

32. Interpolation vs. Approximation Curves
Interpolation Curve – over constrained → lots of (undesirable?) oscillations
Approximation Curve – more reasonable?

33. Blending Functions used to derive the points on curve can be:-
When modelling a curve f(x) by using segments we represent the curve as sum of smaller segments Øi(x) called blending functions.
Blending Functions used to derive the points on curve can be:-
Lagrange Polynomial
Hermite Polynomial
Bernstein Polynomial

34. Spline Representation
What is Spline
Spline Curve
Spline Surface

35. Spline weights are used to make curve without computer
Making Curves
Spline weights are used to make curve without computer

36. Natural Cubic Spline
Spline A long narrow and relatively thin piece or strip of wood, metal, etc.
2 A flexible strip of wood or rubber used by draftsmen in laying out broad sweeping curves, as in railroad work.
Spline Curve is a composite curve formed with polynomial section satisfying specified continuity (smoothness) condition at the boundary.
Spline surface can be described with 2 sets of orthogonal spline curve.

37. Bezier Curve
For n+1 control points pi (0 <= I <= n) the following is the
approximated position vector using Bezier polynomial ftnction
Where B(t)  position vector of curve
Pi  control points
bi,n(t) Bezier blending function
(Bernstein polynomial)

38. Linear Curve (2 control points)
B(t) = (1-t) P0 + t P1
The parameter ‘t’ defines
the path of curve B(t) from P0 to P1

39. Qtadratic Bezier Curve (3 control points)
For the Quadratic Bezier curve B(t) from Q0 to Q1 t varies from 0 to 1.
Q0 varies from p0 to p1
describes a linear Bezier curve.
Q1 varies from p1 to p2
B(t) varies from Q0 to Q1
describes a Quadratic Bezier curve.

40. B(t) = (1-t)2 P0 + 2t (1-t) P1 + t2 P2
Quadratic: Degree 2, Order 3
B(0) = P0, B(1) = P2 B(t) = ? P0 P1 P2
1-t t
Q0(t)
Q1(t)
1-t t
B(t)
B(t) = (1-t)2 P0 + 2t (1-t) P1 + t2 P2

41. Cubic Curve (4 control points)

42. For the Cubic Bezier curve B(t) from R0 to R1 t varies from 0 to 1.
Q0 varies from p0 to p1
describes a linear Bezier curve.
Q1 varies from p1 to p2
Q2 varies from p2 to p3
R0 varies from Q0 to Q1
describes a Quadratic Bezier curve.
R1 varies from Q1 to Q2
B(t) varies from R0 to R1
describes a Cubic Bezier curve.

43. B(t) = (1-t)3 P0 +3t(1-t)2 P1 +3t2(1-t) P2 + t3 P3
Cubic: Degree 3, Order 4
B(0) = P0, B(1) = P2 B(t) = ? P0 P1 P2 P3
1-t t
B(t) = (1-t)3 P0 +3t(1-t)2 P1 +3t2(1-t) P2 + t3 P3 t t
1-t
Q0(t)
Q1(t)
Q2(t) t
1-t t
1-t
R0(t)
R1(t)
1-t t
B(t)

44. For higher order curve

45. Properties of Bezier curve
1. The Bézier curve generally follows the shape of the control polygon, which consists of the segments joining the control points

46. 2. Bézier curves exhibit global control: moving a control point alters the shape of the whole curve.

47. 3. Interpolation. A Bézier curve always interpolates the end control points.s 4. Tangency. The endpoint tangent vectors are parallel to P1- P0 and Pn- Pn Convex hull property. The curve is contained in the convex hull of its defining control points.
Use:- (Due to it’s property & easy implementation)
Available in CAD system
In general graphics package
In drawing & painting package