1. 3D
MODELLING
PART-3
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3. Subdivision Surfaces Provides `Localized Refinement’
Allows you to mix sharp and smooth corners.
Continuity controlled locally.
Increased rendering time
Requires clear understanding to generate good models.
Can be difficult to animate.

4. Parametric Models and Implicit Surfaces
Mathematical representations of complex surfaces.
X^2*Y^3-Z^ X=r(t), Y=r(t)+2*r^2(t), Z=t
Not in production use at this time.
Have very beneficial mathematical properties.
Are extremely difficult to render, modify, and animate.

5. What Kinds of Modeling Operations Exist?
Insert and Delete point, edge, face.
Add Polygon/NURB
Boolean Operations: Union, Intersect, Difference.
Advanced Operations:
Extrude along path.
Loft surface
Revolve
Bend
Split
Very Advanced Operations
Smooth Model

6. What kinds of modeling jobs are there?
General Modelers
Photo-realistic modelers
High poly-count modelers
Low poly-count modelers
Character Animators
Prop Modelers

7. What types of animation controls are available?
Simple – directly keying specific components.
Moderate – An I/K skeleton, Solvers.
Complex – Programmed Mel Scripts, advanced solvers. Linked Solvers.

8. How do you control color and appearance.
Assign Materials.
Assign Shaders.
Assign Textures.
Advanced Shaders – eg Renderman.

9. What about rendering?
Everything is translated to Triangles.

10. How do you Triangulate? Polygons NURBS Subdivision Surfaces
Parametric Surfaces
Implicit Surfaces

11. How is everything rendered.
First everything is converted to polygons
Second everything is converted to triangles.
Then everything is rendered thru the graphics pipeline discussed in the prior lecture.

12. What is Modeling
A process of constructing a virtual 3D graphics object
Modeling tools: creating and constructing complex 3D models fast and easy.
Rendering is a process of creating images from graphics models.

13. A graphics model
geometrical descriptions (particles, vertices, polygons) and associated attributes (colors, shadings, transparencies, materials)
Can be saved in a file using a standard (3D model) file format.

14. Models
Organizational models: hierarchies representing institutional bureaucracies
Quantitative models: equations describing econometric, financial, socialogical, ... systems
Geometric models: collections of components with well-defined geometry and their interconnections
Deformable models: that change forms

15. MODELING CURVES AND SURFACES

16. POLYGON MESHES list of vertices - polygon; list of edges - polygon
list of polygons -- objects
Plane equation from 3 vertices:
Ax + By +Cz + D = 0
Normal: (A,B,C) = k(P1P2 x P1P2)
A, B, and C are proportional to the signed areas
of the projections of the polygon onto the (y, z),
(x, z), and (x, y) planes. If the polygon is parallel
to the (x, y) plane, then A = B = 0.

19. PARAMETRIC BICUBIC SURFACES
General form of cubic curve: Q(u) = U ·M ·G
where G, the geometry vector, is a constant
If we allow G to vary in 3D along some path:
Then, a functional description is often tesselated
to produce a polygon-mesh approximation to the
surface (trianglular polygon patches)
For a fixed t 1
, Q(s, t
) is a curve because G(t
) is
constant. If G i
(t) are cubics, the surface is said
to be a parametric bicubic surface

20. Hermite Surfaces
Curve:
and
Surface:
Since:
we have:

21. Where
Where x coordinates, coordinates of the tangent vectors and twists are specified

22. 1 Just as the Hermite cubic curves, the Hermite bicubic permits C
and G
continuity from one
patch to the next
1st, to have C
continuity, the matching curves
of the two patches must be identical, which
means the control points for the two surfaces
must be identical along the edge
To have C
continuity, the control points along
the edge and the tangent and twist vectors
across the edge be equal.
To have G
continuity, the tangent and twist
vectors across the edge be in the same direction,
but do not need to have the same magnitude. .

23. Bezier Surfaces
The Bezier bicubic formulation can be derived in exactly the same way
as above. The results are:
B-Spline Surfaces
The B-Spline bicubic formulation can be derived in exactly the same
way also. The results are:
Normals to Surfaces
The cross product between the s
and t
tangent vectors of the surface
Q(s, t) results in the normal at given :

24. Solid Modeling Methods (Modeling Solids)u
Solid Modeling Methods (Modeling Solids)

25. Creating solid models. A solid model is defined by volumes.
Hierarchy of entities from low to high: keypoints  lines  areas  volumes.
You cannot delete an entity if a higher- order entity is attached to it.
Areas
Lines &
Keypoints
Keypoints
Lines
Areas
Volumes

26. File formats Representing Solids (solid models)
The domain of representation should be large to allow a useful set of physical objects (solids)
The representation should be unambiguous
Modeling Tools have their own file formats
Volumes
Areas
Lines &
Keypoints
Keypoints
Lines
Areas
Volumes

27. Simple 3D Half-Spaces Sphere Cylinder Cone Torus Box Plane
it splits space into two infinite half-spaces
you can use an infinite cylinder and two planes to make a capped cylinder
You can also get a box from 6 planes…

28. Modeling Approaches Two approaches to creating a solid model:
Top-down
Bottom-up
Top-down modeling starts with a definition of volumes (or areas), which are then combined in some fashion to create the final shape.
add

29. Approaches You may combine both methods.
Bottom-up modeling starts with keypoints, from which you “build up” lines, areas, etc.
You may combine both methods.

30. Top-Down Modeling
Top-down modeling starts with a definition of volumes (or areas), which are then combined in some fashion to create the final shape.
The volumes or areas that you initially define are called primitives.
Primitives are located and oriented with the help of the working plane.
The combinations used to produce the final shape are called Boolean operations.

31. Primitives
2-D primitives include rectangles, circles, triangles, and other polygons.

32. Primitives
3-D primitives: blocks, cylinders, prisms, spheres, and cones.

33. Top-Down Modeling ...Primitives
When you create a 2-D primitive, a modeling tool usually defines an area, along with its underlying lines and keypoints.
When you create a 3-D primitive, a modeling tool usually defines a volume, along with its underlying areas, lines and keypoints.

34. Top-Down Modeling ...Primitives
You can create primitives by specifying their dimensions or by picking locations in the graphics window.

35. Top-Down Modeling Boolean Operations
Boolean operations: combinations of geometric entities: add, subtract, intersect, divide, glue, and overlap, etc.
The “input” to Boolean operations: geometric entities, simple primitives or complicated volumes imported from a CAD system.
add
Boolean operation
Input entities
Output entity(ies)

36. Boolean CSG Operations
Union
Addition, A Ú B
Intersection
A Ù B
Difference
Subtraction, A – B, A Ù not B
Difference is not commutative

37. A more complicated example
Difference of:
Intersection of Sphere and Cube
Union of 3 Cylinders - =

38. Bottom-Up Modeling Most modeling tools use top-down approach
Low level programming systems usually adopts with bottom-up modeling

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