Houdini.School – HS-223 – Maths for Artists with Divyansh Mishra

Info:
This recorded course provides a way for artists to learn and develop an intuition for utilizing mathematical tools in their work using industry-standard programs and methods. This course will not only showcase ways of utilizing math in Houdini but also focus on developing a mindset to solve new problems using math. Students will also learn some math-related tips and tricks to optimize their pre-rendered or real-time projects.

Students will leave this class with a fundamental understanding of different ways in which tools from Mathematics can be utilized for Motion Graphics, Shader Building, and Gameplay Programming. Students will learn to optimize some of their existing setups using Mathematics.

Session 1
Understanding Functions & Number System

In this session, we will develop a solid foundation for the number system and mathematical functions. With the help of real work examples, we will wrap our heads around topics like complex numbers, cartesian & polar coordinates, mapping of mathematical functions in different dimensions, and interpolation with their use cases in Motion Graphics and Shader Building.

Introduction
Number System
Brief Introduction to Complex Numbers
Introduction to Mathematical Functions
Algebra of Functions
Function composition
Use cases of some common functions
Cartesian Coordinates
Developing mindset for the transformation of space using Polar Coordinates
Linear Interpolation & Smooth Step
Group Practice Session
Session 2
Understanding Vectors

In this session, we will learn about vectors. Vectors are one of the most important concepts in computer graphics. They are the building blocks for most of the 2D and 3D motion graphics art pieces. We will discuss different ways in which we can utilize vectors for problem-solving and art direction. We will also briefly look into topics from multivariable calculus to gain insights into vector fields.

Introduction to Vectors
Point Vs Direction
Magnitude & Direction of Vectors
Resolution of Vector
Algebra of Vectors
Dot Product
Cross Product
Vector Fields
Divergence & Curl
Gradient Vectors
Group Practice Session
Session 3
Understanding Quaternions

In this session, we will work with Quaternions and discuss their significance over Euler rotations. Through this session, we will build a relationship between quaternions and complex numbers that can be helpful for debugging our quaternion-based algorithms. We will learn how they can be used to rotate objects in 3D space and help us overcome some of the limitations of matrices.

Introduction to Quaternions
Euler Angles Vs Quaternions
Relationship between Quaternions and Complex Numbers
3D Rotation using Quaternions
Euler Angles to Quaternions
Quaternions to Euler Angles
Quaternion to Rotation Matrix
Quaternion Interpolation
Other useful Quaternion Functions
Group Practice Session
Session 4
Understanding Transformation of Matrices

In this session, we will utilize what we have learned till now to understand the concept of Matrices. We will discuss the need for matrices and their use cases. We will also take a deep dive into the process of packing transformations into a matrix and then unpacking for all sorts of use cases. Overall, we will try to get comfortable with using matrices by understanding them mathematically and conceptually.

Introduction to Matrices
Matrix Multiplication
Unpacking the Matrix
Basis Vectors
Cylindrical & Spherical Coordinate Systems
Transformation using Matrices
Quaternions and Matrices
Matrix Interpolation
Use cases of Matrices in Computer Graphics
Recap