3dareacomputational geometryrectangles
I am having a hard time figuring out if a 3D point lies in a cuboid (like the one in the picture below). I found a lot of examples to check if a point lies inside a rectangle in a 2D space for example this on but none for 3D space.
I have a cuboid in 3D space. This cuboid can be of any size and can have any rotation. I can calculate the vertices $P_1$ to $P_8$ of the cuboid.
Can anyone point me in a direction on how to determine if a point lies inside the cuboid?
If you express the cuboid as $C=\cap_{k=1}^6 H_k$, where each $H_k = \{ x | \langle x, n_k \rangle \le \beta_k \}$, then $ x \in C$ iff $\langle x, n_k \rangle \le \beta_k$ for all $k$.
You also have $ x \in C^\circ$ iff $\langle x, n_k \rangle < \beta_k$ for all $k$.
(It is hard to say if this is the shortest or simplest without more information. The eight vertices or the normals may have some relationship, for example.)
I assume that, as in the question you linked to, the rectangle can have any orientation and is not limited to horizontal/vertical sides.
There are infinitely many rectangles that have two given points as opposite vertices: a third vertex can be any point on the circle that has those two points as the ends of a diameter. A point outside that circle will be in none of the rectangles, while a point on the line segment between the two points will be in all the rectangles. All other points inside the circle but not on the line segment will be in some rectangles and not in others.
Therefore, there is no way to " directly determine if a point lies inside the rectangle without finding the coordinates of the other 2 corners" for those points. Also, there is no way to find the other two corners without additional information.
Best Answer
The three important directions are $u=P_1-P_2$, $v=P_1-P_4$ and $w=P_1-P_5$. They are three perpendicular edges of the rectangular box.
A point $x$ lies within the box when the three following constraints are respected:
EDIT:
If the edges are not perpendicular, you need vectors that are perpendicular to the faces of the box. Using the cross-product, you can obtain them easily:
$$u=(P_1-P_4)\times(P_1-P_5)\\ v=(P_1-P_2)\times(P_1-P_5)\\ w=(P_1-P_2)\times(P_1-P_4)$$ then check the dot-products as before.