# Calculate the four corners of a bounded plane given strike, dip, width, height, and center point?

I am trying to plot a geological fault plane in MATLAB as a bounded rectangle. I want to be able to plot the corners of the bounded fault plane given the following inputs: strike, dip, center point of the plane, and the width and length of the rectangle. A schematic of the problem is shown below:

In the above diagram, $$\alpha$$ is the strike (relative to North), $$\delta$$ is the dip (relative to horizontal), $$P(x_c,y_c,z_c)$$ is the center point of the plane, $$\mathbf{O}$$ is the origin, and $$\mathbf{A}$$, $$\mathbf{B}$$, $$\mathbf{C}$$ and $$\mathbf{D}$$ are the unknown points I want to solve for.

First of all, given the inputs, is it possible to calculate the four corners? If so, what would be the simplest way to do this?

I feel like I could work it out doing a significant amount of trigonometry, but perhaps there is an easier way using vectors that someone already knows which would allow me to avoid re-inventing the wheel.

EDIT: I managed to work it out using trigonometry and came up with the following solution but I still think there is probably a much easier and more elegant way to do it:

alpha = 45; %strike
delta = 30; %dip
w = 4; %width (dipping side)
L = 9; %Length (side parallel to the surface)

A = [0,0,0]; % Pivot point at the surface

H = w*sind(alpha+90);
V = w*cosd(alpha+90);
Z = -w*sind(delta);

B = [L*sind(alpha) L*cosd(alpha) 0]+A;
C = [L*cosd(90-alpha)+H L*sind(90-alpha)+V Z]+A;
D = [w*sind(alpha+90) w*cosd(alpha+90) -w*sind(delta)]+A;

P = [A; B; C; D];

figure(1); p = patch(P(:,1),P(:,2),P(:,3),'red');
xlabel('EW Distance (km)');ylabel('NS Distance (km)');zlabel('Elevation (km b.s.l.)');
axis equal


Thanks.

As a first step suppose $$O$$ coincident with $$A$$, so you can find the unit vectors oriented as the sides $$AB$$ and $$AD$$. That is the vectors: $$\vec u=(\sin \alpha,\cos \alpha, 0)^T \qquad \vec v=(\cos \delta,0,-\sin \delta)$$ so the vector orthogonal to the plane $$ABD$$ is: $$\vec n=\vec u \times \vec v$$ anni, since the plane have to contain the point (correspondent to) $$\vec P$$, its equation is: $$\vec n \cdot (\vec x-\vec P)=0$$ Now, intersect this plane with the coordinate planes $$z=0$$ and $$y=0$$ and you can find the points $$A$$ and $$D$$ and from these the other points.