Fermat point of a triangle

The Fermat point of a triangle is the point that minimizes the total distance to all three vertices. For triangles with all angles less than 120°, this point is unique and has the special property that the lines from it to each vertex form 120° angles.

Here we use gradient descent to find this point by iteratively minimizing the sum of distances to the triangle's vertices.

Setting up the triangle

• create point p0 at (0, 3): \point 0 3
• create point p1 at (5, 0): \point 5 0
• create triangle with vertices p0, p1, p2: t = \triangle p0 p1 p2 - p2 is auto-generated to complete the triangle

These commands set up the trainable point and parameters without changing the visualization:

• create point p at (2, -1): p = \point 2 -1 - this is our initial guess for the Fermat point
• register p as a parameter: \param p - this tells the system that point p is trainable
• create array of triangle vertices: points = \array p0 p1 p2
• set learning rate: learning-rate = \scalar 0.5

Visualizing the connections

• draw lines from p to all vertices: \line p points - visualize the connections

• calculate distances from p to all vertices: dists = \distance p points
• calculate loss as sum of distances: loss = \reduce-sum dists - this is what we want to minimize
• backpropagate the loss: \backprop loss
• do gradient descent: \gradient-descent-step

Click the fermat-train-step command multiple times to continue training and watch point p converge towards the Fermat point. Run the \fermat-point-of-a-triangle command in the geomatic editor to set up the initial state, then repeatedly run \fermat-train-step to optimize.