A corner reflector is modeled as a pyramid in WaveFarer. The following geometric formulas express the relationship between a corner reflector's inner edge lengths and a pyramid's radius and height.
- $a$: length of a corner reflector's inner edge.
- $c = \sqrt{2}a$: length of an edge along the corner reflector's open face.
- $r = \sqrt{\frac{2}{3}}a$: radius of the circle that circumscribes the triangle with three edges of length $c$.
- $h = a/\sqrt{3}$: distance between the center of the corner reflector's open face and its inner vertex.
Background
These equations assume that the corner reflector consists of three faces of equal size. Each face is a triangle with one 90 degree angle and two 135 degree angles. Given an inner length of $a$, the Pythagorean theorem is used to determine the length of an edge along the corner reflector's open face, $c = \sqrt{2}a$. WaveFarer defines a pyramid in terms of its radius and height, so these values' relationship to $a$ and $c$ must be determined.
The pyramid's radius, $r$, defines a circle that circumscribes the equilateral triangle with edges of length $c$. The radius is determined by splitting the equilaterial triangle into three isosceles triangles.
\begin{eqnarray} \cos(30^\circ) &=& \frac{c/2}{r} \\ \frac{\sqrt{3}}{2} &=& \frac{c/2}{r} \\ r &=& \frac{c}{\sqrt{3}} \\ r &=& \sqrt{\frac{2}{3}}a \end{eqnarray}
The pyramid's height, $h$, is the distance between the center of the equilaterial triangle with edges of length $c$ and the corner reflector's inner vertex. The Pythagorean theorem is used to determine $h$ once $r$ has been determined.
\begin{eqnarray} h^2 + r^2 &=& a^2 \\ h^2 + \frac{2a^2}{3} &=& a^2 \\ h^2 &=& \frac{a^2}{3} \\ h &=& a/\sqrt{3} \end{eqnarray}Pyramid Controls
Create a pyramid by right-clicking on Parts in the Project Tree, and selecting Create New ❯ Pyramid to open the pyramid editor.
Three of the editing fields should always use the following settings:
- Top: 0
- Sides: 3
- Create as: Sheet
The remaining fields can be specified as equations, such as in the following examples that assume the corner reflector has an inner edge length of 55 mm:
- Height: (55 mm) / sqrt(3)
- Radius 1: (55 mm) * sqrt(2/3)
- Radius 2: (55 mm) * sqrt(2/3)
A parameter can also be entered into the fields, such as in the following equations that use parameter innerLength set to 55 mm:
- Height: innerLength / sqrt(3)
- Radius 1: innerLength * sqrt(2/3)
- Radius 2: innerLength * sqrt(2/3)
Click Done to create a four-sided pyramid. Then right-click on Pyramid in the Project Tree and select Modify ❯ Remove Faces to create a three-sided corner reflector.