Maths - Projections

This page explains various projections, for instance if we are working in two dimensional space we can calculate:

These transformations are related as we will discuss.

The following pages extend this topic:

To start with the simplest case consider a point projected onto a two dimensional line.

project line

If we want to map the two dimensional space shown here onto the one dimensional space of the line we can use:

P2 = line • P1

where:

In matrix or tensor terms this would be:

P2 =
cos(θ) sin(θ)
P1x
P1y

In other words we transpose the line vector and multiply it by the vector represented by the point in question.

In complex number terms this would be:

P2 = real(line' * P1)

where:

One dimension to a line in two dimensions

We can also do the inverse transform, that is, take an input which is a scalar distance along a line and to map this to a two dimensional position on the line.

P2 =
cos(θ)
sin(θ)
* P1

where:

Result left in 2 dimensions

What if we want to map the points onto a line but leave that line in two dimensional space (as is shown in the diagram above)? We can do this by mapping to a scalar value and then mapping back to two dimensions, that is, apply both of the above transforms in sequence.

We will use the notation:

P2 = P1 || line

That is the component of P1 that is parallel to the line.

In matrix or tensor terms this would be given by:

P2 =
cos(θ)
sin(θ)
cos(θ) sin(θ)
P1x
P1y

Where, in this case,

Multiplying the terms gives:

P2 =
cos2(θ) sin(θ)*cos(θ)
sin(θ)*cos(θ) sin2(θ)
P1x
P1y

We can also calculate the perpendicular distance of the point from the line

We will use the notation:

P2 = P1 perpendicular line

That is the perpendicular distance between P1 and the line.

this is given by:

P2 = (
1 0
0 1
-
cos2(θ) sin(θ)*cos(θ)
sin(θ)*cos(θ) sin2(θ)
)
P1x
P1y

Which gives:

P2 =
1 - cos2(θ) -sin(θ)*cos(θ)
-sin(θ)*cos(θ) 1 - sin2(θ)
P1x
P1y

Using Geometric Algebra Notation

We can repeat the above results using geometric algebra notation.

Mapping 2D to 1D

Mapping a point in 2 (or more) dimensions to a one dimensional (scalar) value to represent the distance along the line.

A first attempt at this would be A•B

where:

This gives a scalar as required, but it is not quite right unless B is unit length, otherwise we can use:

a = (A•B)/|B|

where:

Mapping 1D to 2D

We can use scalar multiplication to get the two dimensional position from the scalar length along the line.

A = (a*B)/|B|

where:

Parallel Component of a Point

We can calculate the parallel component of a point (A) to a line (B) by converting to a one dimensional distance along the line and that converting back to two dimensions. To do this we combine the above expressions as follows:

A || B = A•B * B/|B|2

Perpendicular Component of a Point

A project B = (A x B) x B /|B|2

Summary

Since (B•B) = |B|2 we can rewrite these expressions as:

Higher Dimensions

We can also project onto a line in higher dimensions such as 3D, we just increase the dimension of the vectors that we are using.

Other Projections


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