Triangle's contributory moment of inertia

[Jellyf15h]

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?

 

[Jellyf15h]

Please?

Pretty please? I'll be your friend. You'll get in the credits!

 

[tacman]

Hi there,

Calculating the contributory moment of inertia for a triangle can be quite challenging, but I believe I can help you out. The key to finding the average distance from a point to all points within a triangle is to use the concept of moments of inertia.

First, let's define what we mean by a triangle's contributory moment of inertia. In simple terms, it is a measure of the resistance of a triangle to rotational motion around a specific axis. It takes into account the distribution of mass within the triangle and its distance from the axis of rotation.

To calculate the contributory moment of inertia for a triangle, we need to find the mass moment of inertia for each individual point within the triangle and then sum them up. The mass moment of inertia for a point is calculated by multiplying the mass of the point by the square of its distance from the axis of rotation.

Now, to find the average distance from a point (x,y) to all points within a triangle, we can use the formula for the center of mass of a triangle. This formula takes into account the coordinates of the three points of the triangle and their respective masses. Once you have the coordinates of the center of mass, you can calculate the distance from this point to (x,y) using the distance formula.

Since you mentioned using the three-point formula for area, I assume you already have the coordinates and masses of the three points of the triangle. From there, you can use the center of mass formula to find the coordinates of the center of mass. Then, using the distance formula, you can find the average distance from this point to (x,y).

I hope this helps you in your physics engine design. Let me know if you have any further questions. Good luck!