How to detect whether a circle and a polygon overlap? - math

I am implementing a function that can detect whether a circle and a polygon are overlapping or not.
I have all the points of the polygon and I know the center points and radius of the circle.
There I check two scenarios:
polygon vertices are inside the circle
circle center is inside the polygon
But there are other scenarios in which a circle and a polygon are overlapping, as shown in the attached image. Can anyone suggest validation for finding the intersection?

Here is a possible approach.
If one of the polygon vertices is inside the circle, there is overlap.
If the the circle center is inside the polygon, they overlap. Note that this test is not trivial for non-convex shape. For example consider a polygon similar to a thin spiral.
Otherwise, for every edge (a,b) of the polygon:
Find p, the projection of the circle center to the line (a,b).
If the distance of p to the circle center is larger than the radius, there is no overlap for this edge.
Otherwise, if p is between a and b (so p_x between a_x and b_x, and also p_y between a_y and b_y, to include the case of horizontal and vertical edges), then there is overlap for this edge, otherwise not.

Related

Rotating a polygon so a specific side is on the bottom

I'm trying to find the angle it would take for me to rotate a polygon so that a specific side is completely horizontal and on the bottom.
For example, a shape like this:
Needs to be rotated so the side with the red square on it is on the bottom and completely horizontal, like this:
So far I've tried several approaches but all end up having strange edge cases where the angle is incorrect.
If you have coordinates of two vertices of this edge (x1,y1) and (x2,y2) in counterclockwise order, then rotation angle is
RotAngle = atan2 (y2-y1, x2-x1)

Is there a case where the perimeter of a convex poly is greater than the perimeter of its bounding box?

I know that the area of the polygon is always smaller than the area of its bounding box, but can it have a greater perimeter than the perimeter of its bounding box?
First of all, the area of the polygon may equal that of the bounding box, so I'd say “no larger” instead of “smaller”.
The answer to your question is “No”:
A convex polygon is the intersection of a finite number of half planes, with the additional requirement that the result is compact. So you can obtain every convex polygon by taking its bounding box and cutting away half planes. Each time you cut away a part, you take away two sides of a triangle and replace them by the third. Due to the triangle inequality, this will never increase the perimeter.
In general, if A and B are convex polygons and A is a subset of B, then the perimeter of A is less than or equal to the perimeter of B.

How to calculate the two tangent points to a circle with radius R from two lines given by three points

This is related to the arc drawn by HTML5 canvas "arcTo" function. I need to calculate the two tangent points of a circle with the radius R and two lines given by three points Q(x0,y0), P(x1,y1) and R(x2,y2).
The sketch explains the problem more. I need to find the tangent points A(xa,ya) and B(xb,yb). Note that the center of the circle is not given. Please help.
This is a question of solving a triangle with 2 known angles and one known side. Label the centre of the circle C, then the side you know is BC (or AC if you want). Angle PBC (CAP) is a right angle. The line CP bisects the angle RPQ.
Not all such triangles have a solution.

Get smallest bounding box for a polygon that is large enough despite orientation

I am currently getting the bounding box for my polygon by getting the min/max x and min/max y of the points, but when rotating the polygon the bounding box is too small to fit the rotated polygon. See the illustration for clarification:
This:
Turns into this:
How would I get the bounding box that is big enough to contain any rotated state?
If I understand the problem correctly, this is really trivial.
The point furthest away from the center will always be a vertex. So find the vertex with the maximum distance from the center, and make the box large enough to fit the polygon when that vertex is facing straight up, down, left and right:
Find the vertex furthest away from the center, and let d denote its distance from the center.
The polygon will always fit in the box 2d × 2d.

Math Problem: Getting Coordinates From Position And Angle

let me begin by stating that's i'm dreadful at math.
i'm attempting to reposition and rotate a rectangle. however, i need to rotate the rectangle from a point that is not 0,0 but according to how far its coordinates has shifted. i'm sure that doesn't make much sense, so i've made some sketches to help explain what i need.
the image above shows 3 stages of the red rectangle moving from 0% to 100%. the red rectangle's X and Y coordinates (top left of the red rectangle) only moves a percentage of the blue rectangle's height.
the red rectangle can rotate. focusing only on the middle example ("Distance -50%") from above, where the red rectangle is repositioned at -50 of the blue rectangle's height, its new angle in the above image is now -45º. it has been rotated from its 0, 0 point.
now, my problem is that i want its rotational point to reflect its position.
the red and blue rectangles are the same size, but have opposite widths and heights. since the red rectangle's 0,0 coordinates are now -50% of the blue rectangle's height, and since they have opposite widths and heights, i want the rotational point to be 50% of the red rectangle's width (or 50% of the blue rectangle's height, which is the same thing).
rather than specifically telling the red rectangle to rotate at 50% of its width, in order to do what i want, i need to emulate doing so by using a formula that will position the red rectangle's X and Y coordinates so that its rotational point reflects its position.
Here's an illustrated solution to your problem:
I don't exactly understand what you need, but it seems that a procedure to rotate a rectangle around an arbitrary point may help.
Suppose we want to rotate a point (x,y) d radians around the origin (0,0). The formula for the location of the rotated point is:
x' = x*cos(d) - y*sin(d)
y' = x*sin(d) + y*cos(d)
Now we don't want to rotate around the origin, but around a given point (a,b). What we do is first move the origin to (a,b), then apply the rotation formula above, and then move the origin back to (0,0).
x' = (x-a)*cos(d) - (y-b)*sin(d) + a
y' = (x-a)*sin(d) + (y-b)*cos(d) + b
This is your formula for rotating a point (x,y) d radians around the point (a,b).
For your problem (a,b) would be the point halfway on the right side of the blue rectangle, and (x,y) would be every corner of the red rectangle. The formula gives (x',y') for the coordinates of the corners of rotated red rectangle.
It's quite simple really.
1. Let's settle on your point you want to rotate the rectangle about, i.e. the point of rotation (RP) which does not move when you swivel your rectangle around. Let's assume that the point is represented by the diamond in the figure below.
2. Translate the 4 points so that RP is at (0,0). Suppose the coordinates of that point is (RPx,RPy), therefore subtract all 4 corners of the rectangle by those coordinates.
3. Rotate the points with a rotation matrix (which rotates a point anticlockwise around the origin through some angle which is now the point of rotation thanks to the previous translation):
The following figure shows the rectangle rotated by 45° anticlockwise.
4. Translate the rectangle back (by adding RP to all 4 points):
I assume this is what you want :)
It seems like you could avoid a more complex rotation by more crafty positioning initially? For example, in the last example, position the red box at "-25% Blue Height" and "-25% Red Height" -- if I follow your referencing scheme -- then perform the rotation you want.
If you know the origin O and a point P on the side of rotated rectangle, you can calculate the vector between the two:
(source: equationsheet.com)
You can get the angle between the vector and the x-axis by taking the dot product with this vector:
(source: equationsheet.com)
Given this, you can transform any point on the rectangle by multiplying it by a rotation matrix:
(source: equationsheet.com)

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