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I currently try to flatten a triangle in 3d space while maintaining the edge lengths. The triangle consists of 3 vertices, all with x,y,z coordinates and is drawn clockwise. The second vertex yields 1 for the z value, the other two vertices are aligned to the x-axis and yield 0 for z. Directly setting the z value would violate the edge length constraint. The target is to transform the 3d triangle so it could be completely projected in 2d.

I tried to calculate the angle between a vector lying on the ground and an edge vector to get a rotation matrix. To flatten the triangle I would have to rotate the triangle with exactly this angle. This however is error-prone under real life conditions. I'm currently looking for a way to transform the triangle directly without the need of a rotation.

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  • $\begingroup$ The transformation that you’re describing is precisely a rotation followed by a trivial orthogonal projection onto the $x$-$y$ plane. What’s “error prone” about this? If you don’t want to construct a rotation matrix, you can either use Rodrigues' formula or a pair of reflections, both of which can be done via direct computation on the vertices. $\endgroup$
    – amd
    Commented Jun 29, 2018 at 21:14
  • $\begingroup$ Also, you’re working in 3-D: the vertices are “clockwise” relative to what? $\endgroup$
    – amd
    Commented Jun 29, 2018 at 21:15

2 Answers 2

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Let's say the three vertices of the triangle are $$ \vec{v}_1 = \left [ \begin{matrix} x_1 \\ y_1 \\ z_1 \end{matrix} \right ], \quad \vec{v}_2 = \left [ \begin{matrix} x_2 \\ y_2 \\ z_2 \end{matrix} \right ], \quad \vec{v}_3 = \left [ \begin{matrix} x_3 \\ y_3 \\ z_3 \end{matrix} \right ] $$ You can construct a two-dimensional coordinate system, where $\vec{v}_1$ is at origin, $\vec{v}_2$ is on the positive $x$ axis, and $\vec{v}_3$ is somewhere above the $x$ axis (i.e., has a positive $y$ coordinate).

If we use $$\vec{V}_1 = \left [ \begin{matrix} 0 \\ 0 \end{matrix} \right ], \quad \vec{V}_2 = \left [ \begin{matrix} h \\ 0 \end{matrix} \right ], \quad \vec{V}_3 = \left [ \begin{matrix} i \\ j \end{matrix} \right ]$$ then the rules to keep the edge lengths intact are $$\left\lbrace\begin{aligned} h^2 &= \lVert \vec{v}_2 - \vec{v}_1 \rVert^2 \\ i^2 + j^2 &= \lVert \vec{v}_3 - \vec{v}_1 \rVert^2 \\ (h-i)^2 + j^2 &= \lVert \vec{v}_3 - \vec{v}_2 \rVert^2 \end{aligned}\right.$$ i.e. $$\left\lbrace\begin{aligned} h^2 &= (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 \\ i^2 + j^2 &= (x_3 - x_1)^2 + (y_3 - y_1)^2 + (z_3 - z_1)^2 \\ (h-i)^2 + j^2 &= (x_3 - x_2)^2 + (y_3 - y_2)^2 + (z_3 - z_2)^2 \end{aligned}\right.$$ If the three points are all separate and not on the same line, there is a solution: $$\begin{aligned} h &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \\ i &= \frac{(x_3 - x_1)(x_2 - x_1) + (y_3 - y_1)(y_2 - y_1) + (z_3 - z_1)(z_2 - z_1)}{h} \\ j &= \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2 + (z_3 - z_1)^2 - i^2} \end{aligned}$$ This means that in the triangle, if we drop a line from the third vertex, perpendicular to the line between the first two vertices, the two lines intersect at distance $i$ from the first vertex towards the second vertex. (But note that $i$ can be negative, meaning the opposite direction.) The distance between the third vertex and the intersection is $j$.

OP has a triangle with $z_1 = z_2 = 0$, and wants to find out where the third vertex would be, if the triangle is rotated around the first edge to bring the third vertex to the $xy$ plane.

First, we need two unit vectors. The first unit vector, $\hat{u}$, is from $\vec{v}_1$ towards $\vec{v}_2$: $$\hat{u} = \frac{\vec{v}_2 - \vec{v}_1}{\lVert\vec{v}_2 - \vec{v}_1\rVert} = \left [ \begin{matrix} \frac{x_2 - x_1}{h} \\ \frac{y_2 - y_1}{h} \\ 0 \end{matrix} \right ]$$ The second vector is perpendicular to the first, but also on the $xy$ plane. There are two options: $$\hat{v}_{+} = \left [ \begin{matrix} \frac{y_2 - y_1}{h} \\ \frac{x_1 - x_2}{h} \\ 0 \end{matrix} \right ] \quad \text{or} \quad \hat{v}_{-} = \left [ \begin{matrix} \frac{y_1 - y_2}{h} \\ \frac{x_2 - x_1}{h} \\ 0 \end{matrix} \right ]$$ Typically, you pick the one that is in the same halfspace as $\vec{v}_3$, i.e. has the larger (positive) dot product; this corresponds to the smaller rotation angle: $$\hat{v} = \begin{cases} \hat{v}_{+}, & \hat{v}_{+} \cdot \vec{v}_3 \ge \hat{v}_{-} \cdot \vec{v}_3 \\ \hat{v}_{-}, & \hat{v}_{+} \cdot \vec{v}_3 \lt \hat{v}_{-} \cdot \vec{v}_3 \end{cases}$$

Then, the location of the third vertex on the $xy$ plane is $\vec{v}_3^\prime$, $$\vec{v}_3^\prime = \vec{v}_1 + i \hat{u} + j \hat{v}$$

This is the exact same location you get, if you rotate the triangle around the edge between vertices $\vec{v}_1$ and $\vec{v}_2$, bringing the third vertex also to the $xy$ plane. The two options, $\hat{v}_{+}$ and $\hat{v}_{-}$ correspond to rotations that differ by 180°.

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Given three points $p_1$, $p_2$ and $p_3$.

Define two circles $C_1$ with $p_1$ as center and $l_1$ as radius and $C_2$ with $p_2$ as center and $l_2$ as radius.

Where $l_1 = \|p_1 - p_3\|$ and $l_2 = \|p_2 - p_3\|$ .

The intersection of circles $C_1$ and $C_2$ define two points, one of which can be identified with the point $p_3$.

Now, given any arbitrary plane with normal $n$ containing the points $p_1$ and $p_2$ you can compute the point $p_3$ as one of the intersection points of the above two circles both defined with normal $n$.

See: circle circle intersection + cordinates + 3d + normal plane

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