How do I determine how similar the angles of a triangle are to a given triangle?

Question

I am trying to determine how close a triangle is to a given known triangle based on the angles.

I know the 3 angles of a TRUTH triangle.

I know the 3 angles of an INPUT triangle.

I want to determine a single error value that specifies how close all 3 angles of the INPUT triangle are to the 3 TRUTH triangle angles.

I will evaluate the input triangle against multiple TRUTH triangles to determine which TRUTH the input is closest to.

Answer 1

The question "how close is [this] to [that]" rarely has a single definite answer. It depends on what problem you are trying to solve and what your concerns are. Here is an example. Suppose that the electricity supply to your house is supposed to be constant, and consider the following two scenarios in which it is not what it is supposed to be.

1. The supply is constant, but at a slightly higher level than it should be.

2. The supply is, most of the time, exactly what it should be, but occasionally has a very large but extremely brief increase or decrease.

Which scenario is closer to the "TRUE" supply? Well, if you are concerned about electricity costs, the second is closer, because the increase is extremely brief and won't affect your charges much, whereas the first is constantly making you pay more. On the other hand, if you are concerned about a power surge knocking out your devices, then the first scenario is closer to TRUE because it is never going to do this. So "which is closer" depends on what your concerns are.

So, sorry if this is a non-answer, but you are going to have to decide how you measure the closeness of two triangles, and the decision should be based on what you want to do with the information. Here are two possibilities: MAYBE one of them will work for you. Let the angles of one triangle be $\alpha_1,\alpha_2,\alpha_3$ and the other $\beta_1,\beta_2,\beta_3$, both listed in increasing order.

Option 1: measure the discrepancy between the triangles as the maximum of $|\alpha_1-\beta_1|$, $|\alpha_2-\beta_2|$, $|\alpha_3-\beta_3|$.

Option 2: measure the discrepancy as the sum $|\alpha_1-\beta_1|+|\alpha_2-\beta_2|+|\alpha_3-\beta_3|$. Note that because you know $\alpha_1+\alpha_2+\alpha_3=\beta_1+\beta_2+\beta_3$, it will be possible to simplify this, though the details will depend on which $\alpha$s are bigger than which $\beta$s.

Answer 2

There is no single answer to your question. The answer depends on the definition you choose for the "distance" between two triangles.

If the angles must be matched in a particular order, you can use the sum of the absolute differences between the three pairs of angles, or the sum of the squares of those differences, or the maximum of the differences.

If you are allowed to reorder the vertices you could use the minimum of the six distances computed by either of the methods above.

There are other possibilities. The choice depends on the application you have in mind.