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Graphics Notebook: Maths

This page is part of my “graphics notebook” series, where I keep notes on graphics concepts as I learn about them, to serve as a reference.

Maths
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I’m very much self-taught when it comes to maths, so excuse the obvious definitions, and hopefully there are no errors

Algebra
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  • Use of arithmetic expressions, eg + - / x
  • Use of equations, eg 1 + 2 = 3
  • Maths using letters to represent variables, eg: 2x - y = 9
  • Functions, eg f(x) = 2x

Trigonometry
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  • Maths about relationship between triangle angles and sides
  • Sin, Cos, Tan functions and their inverse
  • SOH CAH TOA , eg Sin(angle) = Opposite / Hypotenuse (angle in radians)
  • Full circle is 2 * PI radians units

Algebra 2
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  • Polynomials arithmetic
  • Complex Numbers (Real & Imaginary) - picture them on a real and imaginary 2D cartesian coordinate system, to map them to a circle.
  • Imaginary number i (i = i, i² = -1, i³ = -i, i⁴ = 1)

Calculus
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  • The maths studying change
  • Differentials (rate of change) & Integration (accumulation)
  • Derivative: instant rate of change at the limit
  • Integral: total accumulated area under a curve
  • Fundamental Theorem of Calculus: Derivative & Integral are each others inverse. (taking the derivative of an integral to it’s limit gives the original function back)

Multi-variable calculus
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  • Same as Calculus but now differentiating or integrating with respect to multiple variables (eg a 2D plane)

Differential equations
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  • Describes how a function changes.
  • Can be taken at increasingly higher orders:
  • The differential of Position is Velocity (1st order)
  • The differential of Velocity is Acceleration (2nd order)
  • The differential of Acceleration is Jerk (3rd order)
  • …

Differential equations come up in fluid simulation & physics effects. For example, when calculating the how much velocity flowed into a cell, we take the integral of the velocity over the distance between two cells. (v / dx)

Linear Algebra
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  • Vectors, Matrices, Transforming Coordinate systems

Geometry
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  • Lines, Shapes, Triangles, Circles, Volume, Area, Perimeter, …

Notation
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  • x : scalar
  • x : vector
  • X : Matrix
  • I : Identity Matrix
  • Σ : Summation (for loop, adding values over X iterations)
  • Π : Product (for loop, multiplying value sover X iterations)

Great resources
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