math for artists

vector fundamental (David Hill slide)

direction

definition	math	case example
		math	C++ example
defining vector \(\overrightarrow{a}\)	$$ \overrightarrow{a} = (\color{red}{a_x}, \color{green}{a_y}, \color{blue}{a_z}) $$	$$ \overrightarrow{a} = (\color{red}{0}, \color{green}{1}, \color{blue}{2}) $$	FVector A = FVector(0.f, 1.f, 2.f);
vector length of \(\overrightarrow{a}\)	$$ ||a|| = \sqrt{\color{red}{a_x}^2 + \color{green}{a_y}^2 + \color{blue}{a_z}^2} $$	$$ ||a|| = \sqrt{\color{red}{0}^2 + \color{green}{1}^2 + \color{blue}{2}^2} $$ $$ ||a|| = \sqrt 5 $$	float ALength; ALength = A.Size(); //
normal / direction of \(\overrightarrow{a}\)	$$ \overrightarrow{n_a} = \frac{\overrightarrow{a}}{||a||} $$	$$ \overrightarrow{n_a} = \frac{(\color{red}{0} , \color{green}{1} , \color{blue}{2})}{\sqrt 5} $$ $$ \overrightarrow{n_a} = (\color{red}{0} , \color{green}{\frac{1}{\sqrt 5}} , \color{blue}{\frac{2}{\sqrt 5}}) $$	FVector ANormal; ANormal = A.GetUnsafeNormal();
normal vector will always have 1 vector length	$$ ||n_a|| = \sqrt{\color{red}{n_x}^2 + \color{green}{n_y}^2 + \color{blue}{n_z}^2} = 1 $$	$$ ||n_a|| = \sqrt{\color{red}{0}^2 + (\color{green}{\frac{1}{\sqrt 5}})^2 + (\color{blue}{\frac{2}{\sqrt 5}})^2 } $$ $$ ||n_a|| = \sqrt{ \color{red}{0} + \color{green}{\frac{1}{5}} + \color{blue}{\frac{4}{5}} } $$ $$ ||n_a|| = 1 $$	ANormal.Size(); // 1.f
so vector contains normal and length	$$ \overrightarrow{a} = (\color{red}{n_x}, \color{green}{n_y}, \color{blue}{n_z}) . ||a||$$

example:

direction and magnitude

$$ \overrightarrow{a} = (\color{red}{a_x}, \color{green}{a_y}, \color{blue}{a_z}) $$ $$ ||a|| = \sqrt{\color{red}{a_x}^2 + \color{green}{a_y}^2 + \color{blue}{a_z}^2} $$ $$ \overrightarrow{n_a} = \frac{\overrightarrow{a}}{||a||} $$ $$ \overrightarrow{n_a} = (\frac{\color{red}{a_x}}{||a||}, \frac{\color{green}{a_y}}{||a||}, \frac{\color{blue}{a_z}}{||a||}) = (\color{red}{n_x}, \color{green}{n_y}, \color{blue}{n_z})$$ $$ ||n_a|| = 1 $$ $$ \overrightarrow{a} = (\color{red}{n_x}, \color{green}{n_y}, \color{blue}{n_z}) ||a||$$ FVector a, n_a;
a = FVector(0.0f, 1.0f, 2.0f);
n_a = a / a.VectorLength();

addition

$$ \overrightarrow{a} = (\color{red}{a_x}, \color{green}{a_y}, \color{blue}{a_z})$$ $$ \overrightarrow{b} = (\color{red}{b_x}, \color{green}{b_y}, \color{blue}{b_z})$$ $$ \overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b} $$ $$ \overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b} = (\color{red}{a_x} + \color{red}{b_x}, \color{green}{a_y} + \color{green}{b_y}, \color{blue}{a_z} + \color{blue}{b_z})$$

example:

substraction

$$ \overrightarrow{a} = (\color{red}{a_x}, \color{green}{a_y}, \color{blue}{a_z})$$ $$ \overrightarrow{b} = (\color{red}{b_x}, \color{green}{b_y}, \color{blue}{b_z})$$ $$ \overrightarrow{c} = \overrightarrow{a} - \overrightarrow{b} $$ $$ \overrightarrow{c} = \overrightarrow{a} - \overrightarrow{b} = (\color{red}{a_x} - \color{red}{b_x}, \color{green}{a_y} - \color{green}{b_y}, \color{blue}{a_z} - \color{blue}{b_z})$$

example:

reflection vector

Wyeth method

dot product trick

$$ ||\overrightarrow{n}|| = 1$$ $$ \overrightarrow{n} = (\color{red}{n_x}, \color{green}{n_y}, \color{blue}{n_z}) $$ $$ \overrightarrow{v} . \overrightarrow{n} = \color{red}{v_x n_x} + \color{green}{v_y n_y} + \color{blue}{v_z n_z}$$ $$ \textrm{length of this side} = \overrightarrow{v} . \overrightarrow{n}$$ $$ (\overrightarrow{v} . \overrightarrow{n}) \overrightarrow{n} $$ tegak lurus di arah normal: $$ \overrightarrow{v} - (\overrightarrow{v} . \overrightarrow{n}) \overrightarrow{n} $$
reflection vector: $$ \overrightarrow{v} - 2 (\overrightarrow{v} . \overrightarrow{n}) \overrightarrow{n} $$

reflection example

transform

transform

visualization

visualization