In this lecture, we'll cover:

• How to define a local element reference frame using an element plane and a chosen global reference plane,
• How to use cross products to obtain the element’s local orthogonal axes,
• How to visualise the element plane, the global plane, and their line of intersection in 3D,

The main idea we work through is how to construct a local coordinate system for an element when its local x-axis is defined by the line of intersection between the element plane and one of the global planes, such as the $xy-$, $xz-$, or $yz-$plane. We first form the element normal as the local z-axis by taking the cross product of two vectors lying in the element plane, then we obtain the local x-axis by taking the cross product of the element normal with the chosen global plane normal. Finally we derive the local y-axis from the cross product of z and x.

We also spend time visualising the geometry so that the construction process is easier to understand. We create a function to draw the chosen global plane in 3D, then add the local axes and, optionally, a highlighted line showing the intersection of the two planes. By switching between different reference planes, we see that the same process produces different local x-axes depending on the chosen global plane.

Next up

In the next lecture, we will calculate the transformation matrix from the local element axes.