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I have three consecutive points of a polygon, say p1, p2, p3. Now I wanted to know whether the orthogonal between p1 and p3 is inside the polygon or outside the polygon.
I do this by taking three vectors v1, v2 and v3. And the point is before the point p1 in the polygon p0.
v1 = (p0 - p1)
v2 = (p2 - p1)
v3 = (p3 - p1)
This polygon is counterclockwise. and It starts from the beginning of v1 and v2.
3 answers
Since your points are consecutive, you can solve this problem by checking the orientation of the triangle p1 p2 p3. If the orientation is the same as a polygon, then the diagonal is inside and outside.
To determine the orientation of the triangle, most in a simple way is to calculate the signed area and check the sign. Compute
P1.x * p2.y + p2.x * p3.y + p3.x * p1.y - p2.x * p1.y - p3.x * p2.y - p1.x * p3.y
If the sign of this value is positive, the orientation is counterclockwise. If the sign is negative, the orientation is clockwise.
To be precise, the method above gives you information about which side of the polygon the diagonal lies on. Obviously, the polygon can still intersect the diagonal at later points.
In principle, the diagonal can be completely inside, completely outside, both inside and outside, and possibly overlap one or more edges in all three cases. This makes it not entirely trivial to determine what you need.
From a mathematical perspective, there really isn't much difference between inside and outside, except for small details like the outside having an infinite area. (At least for a two-dimensional plane, the inside and outside of the playgon do not stand out sharply on the sphere.)
You also have subqueries regarding the ordering of the edges of the polygon. The simplest way is to sum all the angles between adjacent edges in order. This will add up to N*(pi/2). For CCW polygons N is positive.
[edit] Once you know the direction, and if you don't have any of the difficult cases listed above, the question is simple. The angle p0-p1-p2 is less than the angle p0-p1-p3. Therefore, the edge p1-p3 lies at least partially outside the polygon. And if it doesn't intersect the other edge, it obviously lies entirely outside the polygon.
Diagonal in a polygon (polyhedron) - a segment connecting any two non-adjacent vertices, in other words, vertices that do not belong to one side of the polyhedron (one edge of the polyhedron).
For polyhedra a distinction is made between diagonals of faces (considered as flat polygons) and spatial diagonals that extend beyond the boundaries of the faces. Polyhedra with triangular faces have only spatial diagonals.
Counting diagonals
No diagonals for a triangle on the plane and for a tetrahedron in space, since all the vertices of these figures are connected in pairs by sides (edges).
Number of diagonals N for a polygon it is easy to calculate using the formula:
N = n·(n - 3)/2,
Where n- the number of vertices of the polygon. Using this formula it is easy to find that
Number of diagonals of a polyhedron with number of vertices n It’s easy to calculate only for the variant when the same number of edges converge at each vertex of the polyhedron k. Then you can use the formula:
N=n· (n - k - 1)/2,
which gives the total number of spatial and face diagonals. From here it is possible to find that
In this case, the polyhedron converges at different vertices different number ribs, the calculation becomes noticeably more complicated and must be carried out individually for each option.
Shapes with equal diagonals
On a plane There are two regular polygons, which all diagonals are equal among themselves. This square And true pentagon. A square has two similar diagonals that intersect at a right angle in the center. A regular pentagon has 5 similar diagonals, which together form the outline of a five-pointed star (pentagram).
The only true polyhedron with all diagonals are equal between themselves - a faithful octahedron octahedron. It has three diagonals, which intersect in pairs perpendicularly in the center. All diagonals of the octahedron are spatial (the octahedron has no diagonals of faces, since it has triangular faces).
In addition to the octahedron, there is also one true polyhedron, which all spatial diagonals are equal among themselves. This cube (hexahedron). The cube has four similar spatial diagonals, which also intersect in the center. The angle between the cube's digonals is either arccos(1/3) ≈ 70.5° (for a pair of diagonals drawn to adjacent vertices), or arccos(-1/3) ≈ 109.5° (for a pair of diagonals drawn to non-adjacent vertices ).
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