Three Dimensional & Curves and Surfaces

3-D Geometric Primitives

3D geometric primitives are the basic building blocks used to create 3D objects.

Types

1. Points

• Represented as (x, y, z)

2. Lines

• Straight connection between two 3D points

3. Polygons (Faces)

• Usually triangles or quadrilaterals

Common 3D Shapes:

• Cube
• Sphere
• Cylinder
• Cone

3-D Object Representation

Techniques used to represent complex 3D objects in a computer.

Methods:

1. Wireframe Model

• Only edges and vertices

• Simple
• Hard to visualize solid shape

2. Surface Representation

• Defines outer surfaces

Types

• Polygon mesh
• Bézier surfaces

Solid Representation

A. CSG (Constructive Solid Geometry)

• Combine primitives using operations:
  • Union
  • Intersection
  • Difference

B. B-Rep (Boundary Representation)

• Stores surfaces, edges, vertices

3-D Transformations

Operations that change position, size, and orientation of 3D objects.

Types

1. Translation

$$x' = x + t_x,\quad y' = y + t_y,\quad z' = z + t_$$

​2. Scaling 

$$x' = xS_x,\quad y' = yS_y,\quad z' = zS_$$

​3. Rotation

About X-axis

$$y' = y\cos\theta - z\sin\theta$$$$z' = y\sin\theta + z\cos\thet$$

About Y-axis:

$$x' = x\cos\theta + z\sin\theta$$$$z' = -x\sin\theta + z\cos\thet$$

About Z-axis

(similar to 2D rotation)

Matrix Representation

• Uses 4×4 matrices with homogeneous coordinates (x, y, z, 1)

3-D Viewing

Process of displaying a 3D scene from a specific viewpoint.

Components

1. View Reference Point (VRP)

• Position of camera

2. View Plane

• Where image is projected

3. View Volume

• 3D region visible (like a box or frustum)

Steps:

1. Transform to viewing coordinates
2. Clip objects
3. Project onto 2D plane

Projections

Projection converts 3D objects into 2D images.

Types

1. Parallel Projection

A. Orthographic Projection

• No perspective distortion
• Used in engineering drawings

B. Axonometric Projection

• Isometric, dimetric, trimetric

2. Perspective Projection

• Objects appear smaller when far away
• Realistic

Key Difference

Type	Feature
Parallel	No depth effect
Perspective	Realistic depth

3-D Clipping

Removing parts of objects outside the 3D viewing volume.

Clipping Volume

• Defined by:
  • Left, Right
  • Top, Bottom
  • Near, Far planes

Process

1. Check object against all planes
2. Keep visible parts
3. Discard invisible parts

Importance

• Improves performance
• Ensures correct rendering

Final Summary Table

Topic	Key Idea	Example
3D Primitives	Basic shapes	Cube, sphere
Object Representation	Model objects	Wireframe, solid
3D Transformation	Modify objects	Rotate, scale
3D Viewing	Camera view	View volume
Projection	3D → 2D	Perspective
3D Clipping	Remove invisible parts	View frustum

Concept Flow

3D Objects → Representation → Transformation → Viewing → Projection → Clipping → Final Display

Quadric Surfaces

Quadric surfaces are 3D surfaces defined by a second-degree (quadratic) equation:

$$Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$$

Common Types

• Sphere
• Ellipsoid
• Paraboloid
• Hyperboloid
• Cylinder

Importance

• Used in modeling smooth surfaces
• Common in CAD, simulation, graphics

Sphere

$$x^2 + y^2 + z^2 = r^$$

Properties

• All points are at equal distance from center
• Perfectly symmetrical

Applications

• Planets, balls, bubbles
• Lighting models (reflection)

Ellipsoid

Equation:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$

Properties:

• Generalized sphere
• Different radii along axes

Applications

• Modeling irregular objects (eggs, planets)

Blobby Objects (Metaballs)

Blobby objects are smooth, organic shapes formed by combining multiple fields (metaballs).

Key Idea

• Each object contributes a field value
• Surfaces form where values exceed a threshold

Features

• Smooth blending of shapes
• Organic appearance

Applications

• Animation (liquids, smoke)
• Medical imaging
• Special effects

Introduction to Splines

A spline is a smooth curve defined using mathematical functions and control points.

Why Splines?

• Avoid jagged lines
• Create smooth curves

Applications:

• Car body design
• Animation paths
• Fonts and typography

Bézier Curves & Surfaces

A Bézier curve is defined using a set of control points.

Properties

• Starts at first point, ends at last
• Curve influenced by intermediate points

Equation (Cubic): 

$$B(t) = (1-t)^3P_0 + 3t(1-t)^2P_1 + 3t^2(1-t)P_2 + t^3P_3$$

​Advantages

• Easy to control shape
• Smooth curves

Limitations

• Global control (changing one point affects entire curve)

Bézier Surfaces

• Extension of curves into 3D
• Defined using grid of control points

B-Spline Curves & Surfaces

B-Splines (Basis Splines) are more flexible curves defined using

Properties

• Local control (changing one point affects only part)
• Smooth and stable

Advantages

• Better than Bézier for complex shapes
• Efficient for modeling

Disadvantages

• More complex mathematics

B-Spline Surfaces

• Used for smooth surface modeling
• Common in CAD systems

Final Comparison

Feature	Bézier	B-Spline
Control	Global	Local
Flexibility	Less	More
Complexity	Simple	Complex
Use	Simple curves	Complex models

Concept Flow

Quadric Surfaces → Basic Shapes (Sphere, Ellipsoid) → Organic Shapes (Blobby) → Smooth Curves (Splines) → Advanced Curves (Bezier, B-Spline) → Surfaces