Unit 2: Principles of Graphics Modeling

Topics

Simple Geometric Modeling

vertex

edge

face

mesh

Transformations and Modeling

Scene Graphs

Modeling Graphs

Disc.

OpenGL

A way for CPU to communicate with the GPU

Required Libraries and Tooling

GLFW: Graphics Library Framework
GLAD: Loader-Generator

Pipeline

Shape Assembly process creates the primitives

OpenGL doesn’t provide Fragment and Vertex shaders

Coding notes

All OpenGL objects are accessed by the reference

Buffers: In OpenGL app there are two buffers:

Front buffer: which is the displayed frame on the screen
Back buffer: it is used to be filled with colors and graphics in the backend, and it is swapped with the front buffer to make it the main displayed frame

OpenGL doesn’t provide Fragment and Vertex Shaders, so we have to write them and include them in the program

Coordinates and Transformations

There is no absolute positions and coordinate system in the real life

Coordinates: numerical representation of the above objects in a given coordinate system

Points are represented with tuples of coordinates
They are defined to keep tracking the objects
They can be changed for a set of existing objects

Transformation:

Rotation
Scaling
Translation

Objects

Points: represent locations in a space

Vectors: represent movement, force, velocity, and discplacementr from A to B

Normals: represent orientation, and they are vectors with unit legnth

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These are geometric things. Coordinates are the numerical representations in a given coordinate system

Points and vectors are different

The 0 vector means it has no movement and no force
It is meaningful to add vectors, not points

Matrices have two purposes:

Transform things

Rotate an object from facing north to facing east

Change coordinates

Describe the location of the object in relative to another coordinate system

Vectors

A.K.A. Linear Space

Vector space: Set of elements equipped with addition and scalar multiplication

Addition:
Scalar multiplication: where is a scalar (scalar is a real number)

Zero vector: has no displacement or force

In a finite dimensional vector space, given basis vectors , any vector can be written as

refers to the coordinates

is an axis

Linear Algebra Notation

Can be written as:

Shorthand:

refers to the coordinates

Linear Transformations/Maps

Transformation L of the vector space so that:

We use to say: “maps to”

Matrix Notation

L is called the Linear Maps and it is determined by:

Algebraic Notation of Linear Map

The action of L leaves the coordinates alone but acts on the basis vectors

are vectors in the same space and expressed as follows:

So, we get this for 3 basis vectors:

By combining all of them together:

Thus, the notation of linear map:

B: matrix of basis vectors
c: coordinates
M: coordinates of images of all vectors all under the linear map (the transformation matrix)

There are two interpretations of linear map :

Transforming the basis
Transforming the coordinates

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The change of basis vector is critical in computer graphics, computer vision, and robotics

Change of Basis

Think of matrix multiplication is a piece of code that:

takes a vector
and output the matrix times the vectors

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By convention, matrix multiplication is done from right to left

Vectors can be expressed in a basis

We change the basis to find the coordinates of some vector in a different coordinate system

Assume we have the two bases and and we have the coordinates of in

Then

Now, given the coordinate of in B: , we can find the coordinates of in :

Types of transformations (not all of them are linear):

Translation is not linear. You cannot translate vectors.

Points

Point is a location

Adding points is not meaningful

Boston location + NY Location = ?!

Vector is a motion between two points

Adding vectors is meaningful

3km north + 4km east = 5km north-east

Affine Space

Points are elements of an affine space

Affine spaces are an extension of vector space

We denote them with a tilde

Algebra Notation

Points can be expressed in a frame (origin + basis)

is the origin
The extra dummy coordinate is always 1 (for now to preserve the origin information)

Point-Vector Operations

(displacement) Point - Point:
Point + Vector:

Affine Transformation

All transformation are allowed including the translation

Applying an affine transform to point is similar to vector. We just add an additional dimension.

Linear Transformation (Linear Map )

In this kind of transform, we leave the fourth component of the map matrix without changing it.
That’s because we don’t want to change the origin

Translation Component (Translation Map)

In this process, we add a translation vector to the point(s)

This is the vector in the basis

Remember that:

Then, the full translation expression is:

Since we care about changing the origin, we keep the information of the latest dimension in the transformation matrix

The generic affine transformation expression is:

Notation Props

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These props are useful to keep tracking the information about the objects

If the fourth coord is zero, then we have a vector (like in subtraction)

If the fourth coord is one, then we have a point (like in addition)

Vectors are not affected by the translation part

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The additional coordinate is useful to distinguish between vectors and points: - In 3D space → We add a 4th dimension - In the 2D plane → We add a 3th dimension