Curve, Surface and Solid

1. CHARACTERISTICS OF BEZIER
CURVE
The curve interpolates the first and last control points.
The curve is tangent to the first and last segments of
characteristics polygon.
The curve shape can be modified by either changing one
or more vertices of its polygon or by keeping the polygon
fixed and specifying multiple coincident points at a vertex
as shown in figure.

4. B-SPLINE CURVE
B-spline curves provides another effective method, besides
Bezier, of generating curves defined by data points.
In fact, B-spline curves are the proper and powerful
generalization of Bezier curves.

5. COX-DEBOOR RECURSION FORMULA

6. THE B-SPLINE FUNCTIONS HAVE
FOLLOWING PROPERTIES

7. CHARACTERISTICS OF B-SPLINE
CURVES

10. RATIONAL CURVES
 A rational curve is defined by the algebraic ratio of two polynomials
while a nomenclature curve is defined by one polynomial.
 A rational B – spline curve defined by
𝑃 𝑢 =
𝑖=0
𝑛
𝑃𝑖 𝑅𝑖 𝑘 𝑢 , 0 ≤ 𝑢 ≤ 𝑢 𝑚𝑎𝑥
 𝑅𝑖, 𝑘 𝑆 are the rational B – spline Basis function are given by
𝑅𝑖 𝑘 𝑢 =
ℎ𝑖 𝑁𝑖 𝑘 (𝑢)
𝑖=0
𝑛
ℎ𝑖 𝑁𝑖 𝑘 𝑢

12. GEOMETRIC MODELING
The mathematical description of the geometry of an object
using a software is called as geometric modeling
There are three basic methods
Wire – Frame Modeling
Surface modeling
Solid modeling

13. WIRE FRAME MODELING
This is one of the most popular and commonly used
method of geometric modeling.
In construction of wire frame model, the edges of an
object are presented as lines.
Wire frame model is used for following representations
2D Representation
Orthographic views representation

14. 2D Wire Frame Model 3D Wire Frame Model
The co-ordinate system is 2D co-ordinate
system i.e. x and y co-ordinates only
3D co-ordinate system is used for
representing objects; x, y and z
coordinates are used
3 Dimensional wire frame system
generation is difficult
Both 2D and 3D wire frame
generation is possible
Hidden lines may not complicate the
figure
Difficult for the viewer to interpret the
figure unless the hidden lines are
removed
Curved surfaces are indicated by circles,
arcs and ellipses
Curved surfaces are represented by
spaced generators.

15. DRAWBACK OF WIRE FRAME MODEL
Lack of clarity
When part geometry is complex and in case of 3D wire
frame systems, the hidden lines causes the image to be
confusing to the viewer.
During the surface definition, there might be confusing.
By eliminating hidden lines, some clarity can be obtained.

16. SURFACE MODELING
The techniques of representation of objects (or) components by
surface is called surface modeling.
Objects can be clearly interpreted by the user.
Main draw back here is that, no data is available about the
interior of solid.
Application is modeling car bodies, ships, aerospace structure,
dies, etc.

17. TECHNIQUES AVAILABLE FOR SURFACE
MODELING
Surface patch
Coons patch
Bicubic patch
Bezier surfaces
B-spline surfaces
NURBS (Non-Uniform Rational B-Splines)

18. SURFACE PATCH
The building blocks of the surfaces are known as surface patch.
Generally u and v are two variables used for representing a patch.
0 ≤ 𝑈 ≤ 1 𝑎𝑛𝑑 0 ≤ 𝑉 ≤ 1

20. COONS PATCH
Coons surface batch is a form of “transfinite interpolation”,
which indicates that the coons scheme interpolates to an
infinite number of data points.
Coons patch is particularly useful in blending four prescribed
intersecting curves which forms a closed boundary.
In practice, coons surface can be used to create windshield
designs for cars and skyline windows for houses.

22. BEZIER SURFACE
Bezier surface is an extension of the Bezier curve in two parametric
directions u and v.
An orderly set of data or control points is used to build a topologically
rectangular surface as shown in figure.
The surface equation can be written as
𝑃 𝑢, 𝑣 =
𝑖=0
𝑛
𝑗=0
𝑚
𝑃𝑖𝑗 𝐵𝑖,𝑛 𝑢 𝐵𝑗,𝑚 𝑣 , 0 ≤ 𝑢 ≤ 1, 0 ≤ 𝑣 ≤ 1
where,P(u,v) is any point on the surface
𝑃𝑖𝑗 are the control points

24. B-SPLINE SURFACE
B - Spline surface is an extension of the B - Spline curve. A
rectangle set of data points creates the surface.
A B – Spline surface can approximate or interpolate the
vertices of the polyhedron as shown in figure.
B – Spline surface equation is defined by
𝑃 𝑢, 𝑣 =
𝑖=0
𝑛
𝑗=0
𝑚
𝑃𝑖𝑗 𝑁𝑖,𝑘 𝑢 𝑁𝑗,𝑙 𝑣 , 0 ≤ 𝑢 ≤ 1, 0 ≤ 𝑣 ≤ 1
where,P(u,v) is any point on the surface
𝑃𝑖𝑗 are the control points

27. SOLID MODELING
Solid modeling is one of the most effective geometric
modeling method. In this approach, models are displayed as
solids to viewer, there by eliminating any chance of
misinterpretation.
In solid modeling , to make the object more realistic.

32. SOLID MODELING TECHNIQUES
 Half – space Method
 Boundary representation Method
 Constructive Solid Geometry (CSG)
 Analytical Solid Geometry (ASM)
 Primitive instancing
 Sweep representation
 Spatial portioning representation

33. CONSTRUCTIVE SOLID GEOMETRY (CSG)
This method is also known as C-rep. In this method, solid
graphic primitives are employed for constructing the model.
The solid primitives include cubes, spheres, cylinders,
rectangle blocks and pyramids.

37. CSG MODEL
The constructive solid model uses building block approach
The physical objects can be divided into set of elements and
combined in order to form an object.

38. ADVANTAGES OF CSG
This requires less storage space
This method is advantageous in the initial creation of solid
models. Using Boolean operations, it is easy to construct solid
models precisely.
Less skill is enough
CSG is more user friendly

39. DISADVANTAGES OF CSG
This method involves more computational work for creating a
solid model
For complicated solid geometry, in this method is not
appropriate
The tree is not unique for the same part design

40. Boundary representation Method
Boundary representation is one of the most popular and widely
used schemes to create solid models of physical objects.
In this method, front view, top view, bottom view, side view of
an object is sketched and connected by means of lines to create
a relationship.

41. PRIMITIVES OF B-rep MODEL
Edge
Vertex
Face
Loop
Genus or Handle
Body

42. POLYHEDRAL OBJECT
Simple polyhedra
Polyhedra with faces of inner loops
Polyhedra with no through hole
Polyhedra with through holes

44. EULER EQUATIONS
 For simple polyhedra
V – E + F = 2
 For closed polyhedra object
F – E + V – L = 2 (B - G)
 For open polyhedra object
F – E + V – L = B - G

47. ADVANTAGES OF B-rep
This method is very powerful for creating complex shapes
solid models.
B-rep model can be easily converted into wire frame model
system.
B-rep system stores an explicit definition of the model
boundaries.
B-rep system is very much compatible with other systems.

48. DISADVANTAGES OF B-rep
This requires more storage space.
This concept cannot be applied for tool path generation.