Solving Linear Interpolation for 3-D Triangle Edges

[laminatedevildoll]

I have a 3-D traingle, and the edges are a,b,c. If I want to find a line interpolation at a point in the center, let's say that it's P(x,y)...

My equations are
$$\Delta Z$$ = A + Bx + Cy
$$\Delta Z_a$$ = A + Bx_a + Cy_a
$$\Delta Z_b$$ = A + Bx_b + Cy_b
$$\Delta Z_c$$ = A + Bx_c + Cy_c

In order to solve for $$\Delta Z$$, how do I use the above equations? Do I have to add them (equations 2,3,4) all up and substitute in A for the first equation?

To find the coefficients, do I just solve for A,B,C after I know what $$\Delta Z$$ is?

I also have to find out P(y) by assuming that I know what P(x,z) is. For this do I just substiture y for z to the same equations?

Also, I have to find a best average planar rectangle from this.

I'd appreciate any help.

 

[pat_connell]

Yes, you can solve for \Delta Z by adding the equations (2,3,4) together and substituting in A for the first equation. To find the coefficients, you would solve the system of equations for A, B, and C. To find P(y), you would substitute y for z in the same equations. For the best average planar rectangle, you would need to calculate the area of the triangle, then divide that by the area of the plane, and then use the result to calculate the coordinates of the rectangle (which would be the center point, width, and height).

 

[M98Ranger]

To solve for the linear interpolation at point P(x,y), you can use the above equations by substituting the values for x and y into the equations and solving for \Delta Z. This will give you the interpolated value for the height at point P.

To find the coefficients A, B, C, you can use the method of least squares. This involves finding the values of A, B, and C that minimize the sum of the squared errors between the actual values of \Delta Z and the interpolated values at points a, b, and c. Once you have these coefficients, you can use them to find the interpolated value at point P.

To find P(y), you can substitute the value of y into the equations and solve for \Delta Z. This will give you the interpolated value for the height at point P along the y-axis.

To find the best average planar rectangle, you can use the interpolated values at points a, b, and c to define a plane and then find the best fitting rectangle within that plane. This can be done using techniques such as least squares or geometric methods.

I hope this helps! Let me know if you have any further questions.