Transforming Triangle in 3 Space with z = 1-x-y

In summary, The conversation discusses a transformation of a triangle in 3D space with a constraint on the z-coordinate. The transformation matrix A and three vector equations are provided, along with the vertices of both the original and transformed triangles. The issue is that the description is not easily solvable without additional clarification. It is mentioned that the transformed points can stretch and translate, but not rotate. The problem is then constrained further to generate n equations in n unknowns, and the speaker has found a solution for three of the unknowns. There are now only 6 equations left to solve.
  • #1
albert281
13
0
I am trying to transform a triangle in 3 space with the constraint that z = 1-x-y. At this point I have the following transformation matrix A, and three vector equations:

| a1 a2 a3 |
A = | a4 a5 a6 |
| a7 a8 a9 |



A*x1 = k1*y1
A*x2 = k2*y2
A*x3 = k3*y3




I know the vertices of both triangles, before and after the transformation. The problem is that I can't put the above description in a manner that is solveable...without some input to clarify the issue.

The transformed triangle points can move in a manner to suggest stretching and translation, but not rotation. Given that z is dependent on x and y, how can I constrain this problem such that I can generate n equations in n unknowns?
 
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  • #2
I figured out how to solve for a7, a8 and a9 by using the constraint x+y+z=1 and solving for 3 equations in 3 unknowns. I found k1=k2=k3=1. I only have 6 equations in 6 unknowns left! Yay!
 

1. What is a transforming triangle in 3 space with z = 1-x-y?

A transforming triangle in 3 space with z = 1-x-y is a mathematical concept where a triangle is placed in a three-dimensional coordinate system with the z-value being defined by the equation 1-x-y. This means that the z-coordinate of each point on the triangle will change depending on the values of x and y, resulting in a transformation of the triangle in 3D space.

2. How is the transformation of the triangle determined?

The transformation of the triangle is determined by the equation z = 1-x-y, which is used to calculate the z-coordinate of each point on the triangle. The x and y values are the coordinates of the point on the triangle, and when substituted into the equation, the resulting z-value will determine the position of the point in 3D space.

3. What does the transformation of the triangle represent?

The transformation of the triangle represents the changing z-coordinate of each point on the triangle, resulting in a 3D shape that is distorted or stretched in certain areas. This can be visualized as the triangle being pulled or pushed in different directions, depending on the values of x and y.

4. How is the transformation affected by different values of x and y?

The transformation is affected by different values of x and y because they determine the z-coordinate of each point on the triangle. Changing the values of x and y will result in a different z-value, which will in turn affect the position of the point in 3D space and alter the overall shape of the triangle.

5. What are the practical applications of a transforming triangle in 3 space with z = 1-x-y?

A transforming triangle in 3 space with z = 1-x-y has many practical applications in fields such as computer graphics, engineering, and physics. It can be used to visualize and analyze 3D shapes and transformations, as well as model real-world phenomena such as fluid flow, heat transfer, and structural deformations.

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