Graphics and projective geometry: equation reference

Sometimes I do some GLSL or similar: much graphics and geometry. I usually scribble down some back-of-envelope variant of the usual geometry equations, but I’ve never consistently decided what to call things or how to write them down.

Here I’m setting down the notation I’m going to use, and capturing the definitions. Reference pseudo-implementations are given too (Swift).

The equations and source given here have been unit tested; full source to come after tidying.

Glossary

• scalar absolute value: \( | \mathbf{a} | \)
• vector length:¹ \( \| \mathbf{a} \| = \sqrt{ \mathbf{a}_0^2 + \mathbf{a}_1^2 + \dots } \)
• vector length squared: \( \| \mathbf{a} \|^2 = \mathbf{a}_0^2 + \mathbf{a}_1^2 + \dots \) (also equal to \( \mathbf{a} \cdot \mathbf{a} \))
• unit vector: \( \hat{\mathbf{a}} = \mathbf{a} / \| \mathbf{a} \| \). For larger expressions can write as \( unit( \dots ) \)
• angle between vectors: written as \(\angle (\mathbf{a}, \mathbf{b})\)
• orthogonal vectors: \(\angle (\mathbf{a}, \mathbf{b}) = 90^\circ\). Can be asserted with \(\mathbf{a} \perp \mathbf{b}\)
• orthogonal vector: in 2D, a vector \(\mathbf{a}\) has its orthogonal vector written as \(\mathbf{a}^{\perp}\). There are two possible orth vectors, we define \(\mathbf{a}^{\perp} = \operatorname{rot}_{90} \mathbf{a} \) (i.e. \(\mathbf{a}\) rotated 90 degrees counter-clockwise). The other orth vector is \(\mathbf{a}^{\top} = \operatorname{rot}_{-90} \mathbf{a} = -\mathbf{a}^{\perp} \)
• same direction vectors: \(\angle (\mathbf{a}, \mathbf{b}) < 90^\circ\).
• opposite direction vectors: \(\angle (\mathbf{a}, \mathbf{b}) > 90^\circ\).
• dot product: a scalar of \(\mathbf{a}\) and \(\mathbf{b}\), written \(\mathbf{a} \cdot \mathbf{b}\), computed as sum of all components of two vectors multiplied together. Is equal to \(|\mathbf{a}| |\mathbf{b}| \cos \theta\)
• determinant: written \(\det[\mathbf{a}, \mathbf{b}]\). In 2D defined as \(a_0 b_1 - a_1 b_0\). Is equal to \(|\mathbf{a}| |\mathbf{b}| \sin \theta\) and also \(\mathbf{a} \cdot \mathbf{b^{\perp}}\)

Projection length

The length of the projection of \(\mathbf{a}\) onto \(\mathbf{b}\).

func projectionLength(a: Vec2, b: Vec2) -> Double {
        let bUnit = b.unitVector
        return a.dot(bUnit)
}

Projection

The projection of \(\mathbf{a}\) onto \(\mathbf{b}\), which can be in the same direction or opposite direction to \(\mathbf{b}\), or zero.

func projection(a: Vec2, b: Vec2) -> Vec2 {
        let bUnit = b.unitVector
        return a.dot(bUnit) * bUnit
}

Forward projection

The projection of \(\mathbf{a}\) onto \(\mathbf{b}\). It can be in the same direction as \(\mathbf{b}\) or zero.

func projectionForward(a: Vec2, b: Vec2) -> Vec2 {
    let bUnit = b.unitVector
    return abs(a.dot(bUnit)) * bUnit
}

Rejection length (2D only)

The length of the rejection of \(\mathbf{a}\) onto \(\mathbf{b}^\perp\).

func rejectionLength(a: Vec2, b: Vec2) -> Double {
    let bUnitOrth = b.unitVector.rotated90CCW
    return a.dot(bUnitOrth)
}

Rejection (2D only)

The rejection of \(\mathbf{a}\) onto \(\mathbf{b}\), which is defined as the projection of \(\mathbf{a}\) onto \(\mathbf{b}^\perp\).

func rejection(a: Vec2, b: Vec2) -> Vec2 {
    let bUnitOrth = b.unitVector.rotated90CCW
    return a.dot(bUnitOrth) * bUnitOrth
}

Right rejection (2D only)

The projection of \(\mathbf{a}\) onto \(\mathbf{b}^\perp\), fixed to the right of \(\mathbf{b}\).

func rejection(a: Vec2, b: Vec2) -> Vec2 {
    let bUnitOrth = b.unitVector.rotated90CCW
    return abs(a.dot(bUnitOrth)) * bUnitOrth
}

Rejection (dimensions >= 2)

This equation also works for 2D, but if you’re specifically working in 2 dimensions use the simpler 2D version above.

Is same direction

This equation² returns 1 (true) if \(\angle (\mathbf{a}, \mathbf{b}) > 0\).

func isSameDirection(a: Vec2, b: Vec2) -> Bool {
    a.dot(b) > 0
}

Is opposite direction

This equation returns 1 (true) if \(\angle (\mathbf{a}, \mathbf{b}) > 0\).

func isOppositeDirection(a: Vec2, b: Vec2) -> Bool {
    a.dot(b) < 0
}

Is to left (2D)

/// is A to left of B
func isToLeft(a: Vec2, b: Vec2) -> Bool {
    a.rotated90CCW.dot(b) < 0
}

Is to right (2D)

/// is A to right of B
func isToRight(a: Vec2, b: Vec2) -> Bool {
    a.rotated90CCW.dot(b) > 0
}

Rebase (2D)

Converts a 2D vector into a different basis with \(\mathbf{b}\) being the first (\(x\)) base vector in new basis:

Summary

The Ref output is the result when applying the function to these values:

Think of it as a single-value unit test for validating an implementation of the funcs.

Note that any function using the orth operator (\( \perp \)) applies to 2D only.

Alternative to subscripts for bigger expressions

If you’re writing an operator and you have some unwieldy expression for the \( \mathbf{b} \) subscript, you can use brackets as an alternative form:

Notice that it’s the same order (\( \mathbf{b} \) then \( \mathbf{a} \)) as the \( \operatorname{proj}_{\mathbf{b}} \mathbf{a} \), to avoid confusion.

Applications of cos and sin

Cos is associated with deciding direction (forward versus back). This is because its zero points are at -90 and 90 degrees; it has unique values³ from 0 to 180 degrees.

Sin is associated with deciding handedness/side (left versus right). This is because its zero points are at 0 and 180 degrees; it has unique values from -90 to 90 degrees.

• Graphics
• Dev
• Maths
• Geometry