Projecting 3D on 2D

A point in a three dimensional coordinate system can be represented by a \(3\times 1\) matrix. When modelling three dimensions on a two dimensional computer screen, you must project each point to 2D. After the projection, each point is represented by a \(2\times 1\) matrix.

If we assume that the z-axes (or the third base vector) is pointing from the computer screen, then the screen is a projection on the xy-plane.

The projection matrix \(\mathbf{P}\) must map the matrix \(\mathbf{v}=\left( \begin{array}{} a\\b\\c\\ \end{array}\right)\) onto a matrix \(\mathbf{w}=\left( \begin{array}{} a\\b\\ \end{array}\right)\). This can be achieved by letting

\[\mathbf{P}=\left( \begin{array}{} 1 & 0 & 0\\0 & 1 & 0\\ \end{array}\right)\]

Then let \(\mathbf{w}=\mathbf{P}\mathbf{v}\).

Matrix as a visual point

In GeoGebra, matrices are represented by lists. A \(2\times 1\) matrix is not shown in the drawing pad.

If you want to show a matrix \(\mathbf{matrix1}=\binom{2}{1}\) as a point, you must use the matrix elements as coordinates for a new point. You retrieve an element of a matrix by using the command Element[<matrix>, <row>, <column>].

To show a \(2\times 1\)-matrix \(\mathbf{w}\) as a point \(A\), you write: A=(Element[w,1,1],Element[w,2,1])

Exercise

• Make three sliders to represent the matrix elements \(v_x, v_y, v_z\) of a matrix \(\mathbf{v}\).
• Create the matrix \(\mathbf{v}\) by using the spreadsheet or by entering it as a list in the input bar: v={{v_x},{v_y},{v_z}}
• Create the projection matrix \(\mathbf{P}\).
• Create the projected matrix \(\mathbf{w}=\mathbf{P}\mathbf{v}\) in the input bar: w=P v
• Create a point A, having the elements of \(\mathbf{w}\) as coordinates.
• Change the sliders. When changing the \(v_z\)-slider, nothing should happen with the point A.

This continues with the rotations on the next page.