Triangle's contributory moment of inertia

AI Thread Summary
The discussion focuses on finding an algorithm to calculate the average distance from a point to all points within a triangle, which is essential for a 2D physics engine's rotational inertia calculations. The user seeks a solution that avoids complex calculus and prefers a straightforward algorithm. They mention using a three-point formula for area and a trigonometric method for determining the mass midpoint of triangles, but express a desire for alternatives that are computationally simpler. The goal is to integrate these calculations for solid shapes composed of multiple triangles to accurately compute their center of rotation. The request emphasizes the need for practical solutions in game and simulation design.
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Triangle's contributory moment of inertia :D

Howdy.

I'm designing a 2-dimensional physics engine for use in games and simulations and have hit a roadblock.
The rotation system requires a value corresponding to the proportion between an object's regular inertia and its rotational inertia. EG, for a ring the proportion is 1, for a disc it's .5 evidently. Basically, <THIS> is what I'm after.

Anyway, what I need is an algorithm to find the average distance [not scaled] from a given point (x,y) to all points in the AREA of a triangle [(x1,y1), (x2,y2), (x3,y3)]. To say it differently, I need to find the average distance from all points within a given triangle to (x,y). Being in basic calculus [CURSE YOU PUBLIC EDUCATIONNNN] I'm not so familiar with integrals, so a solved algorithm is what I need.

Just to provide extra perspective, solid shapes will consist of multiple triangles [as it pertains to area, anyway.] and a weighted mean based on area will be used when computing their overall center of rotation and their average distribution. [what we're finding.] I plan to use the three-point formula for area and a trig algorithm to find the mass midpoint of each triangle. [If there's a way to do this without trig, tell me. Those tend to run faster.] There will be no variation in density in a triangle.

So, who among you is man enough to crack this nut?
 
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