Translations

Consider the following 3D triangle,

$$((x_1,y_1,z_1), (x_2,y_2,z_2), (x_3,y_3,z_3)) ,$$ (3.2)

in which its vertex coordinates are expressed as generic constants.

Let $x_t$, $y_t$, and $z_t$ be the amount we would like to change the triangle's position, along the $x$, $y$, and $z$ axes, respectively. The operation of changing position is called translation, and it is given by

$$\begin{array}{l} (x_1,y_1,z_1) \mapsto (x_1+x_t, y_1+y_t, z_1... ...(x_3,y_3,z_3) \mapsto (x_3+x_t, y_3+y_t, z_3+z_t) , \end{array}$$ (3.3)

in which $a \mapsto b$ denotes that $a$ becomes replaced by $b$ after the transformation is applied. Applying (3.3) to every triangle in a model will translate all of it to the desired location. If the triangles are arranged in a mesh, then it is sufficient to apply the transformation to the vertices alone. All of the triangles will retain their size and shape.

Figure 3.4: Every transformation has two possible interpretations, even though the math is the same. Here is a 2D example, in which a triangle is defined in (a). We could translate the triangle by $x_t= -8$ and $y_t= -7$ to obtain the result in (b). If we instead wanted to hold the triangle fixed but move the origin up by $8$ in the $x$ direction and $7$ in the $y$ direction, then the coordinates of the triangle vertices change the exact same way, as shown in (c).