DarkBasic Code Corner

Intersection of 2 ellipses: <Example>
Easiest way to test for this intersection is to convert the ellipses into circles. Convert one ellipse into a unit circle (circle with a radius of 1) then convert the other ellipse using the same transformation.
A - ellipse A
A.x, A.y - ellipse center
A.rx, A.ry - radii of ellipse on X and Y axis, respectively
B = ellipse B
tA = transformed ellipse A
tB = transformed ellipse B

tA.x = A.x / A.rx
tA.y = A.y / A.ry
tB.x = B.x / A.rx
tB.y = B.y / A.ry
tB.rx = B.rx / A.rx
tB.ry = B.ry / A.ry

Now find the point on ellipse B nearest to the unit circle(ellipse A)

angle# = atanfull(tA.x-tB.x, tA.y-tB.y)
x# = tB.x + sin(angle#)*tB.rx
y# = tB.y + cos(angle#)*tB.ry

Finally, check the distance. If the center of transformed ellipse A and point [x,y] is less than or equal to 1 then the ellipses intersect.
if (tA.x-x#)^2 + (tA.y-y#)^2 <= 1.0 then TRUE
Intersection of 2 line segments:
Finds the intersection time t of line[A,B] through line[C,D]. If line segments intersect, time value will be in set [0,1]. If value is outside of this range, the line segments do not intersect.

n = [(D.x-C.x) * (A.y-C.y)] - [(D.y-C.y) * (A.x-C.x)]
d = [(D.y-C.y) * (B.x-A.x)] - [(D.x-C.x) * (B.y-A.y)]
t = n / d

To find the point of intersection, I:

I.x = A.x + (B.x - A.x)*t
I.y = A.y + (B.y - A.y)*t
Intersection of a circle and a line segment:
To determine if a line and a circle intersect, simply calculate the distance from the circle's center to the line. If the distance is less than or equal to the circle's radius, then the two intersect.
Intersection of an ellipse and a line segment:
Nearly identical to the above method, but with an extra step. Transform the ellipse into a circle and transform the remaining coordinates used in the problem into the same space as used to transform the ellipse.
Given ellipse E and line L with endpoints [A,B] and [C,D], transform the ellipse into circle C with a radius of 1:

L.ax = L.ax / E.rx
L.ay = L.ay / E.ry
L.bx = L.bx / E.rx
L.by = L.by / E.ry
C.x = E.x / E.rx
C.y = E.y / E.ry
C.r = 1

Now find the distance from point [C.x,C.y] from the transformed line L. If the distance is less than or equal to 1 then the ellipse and line intersect.
Point in a capsule:
Another simple problem solved by finding the distance of a point from a line. Assuming you can define your capsule using the endpoints of a line segment [A,B] and radius r:

If the distance of point P from line segment [A,B] is less than or equal to the radius r of the capsule, then the point is inside the boundaries of the capsule.

Note: If you would like to check between a circle rather than a point, use the circle's center coordinates as point P and find the distance from the line making up the capsule. If the distance is less than the combined size of the capsule's radius and the circle's radius, the circle and capsule intersect. Using an ellipse instead only requires you to convert the ellipse into a circle with a radius of 1 and tranforming the capsule according to the same spacial proportions.

Point in a 2D triangle:

Determine if point P is inside triangle[A,B,C].
First, you need to know the order the vertices are in. Next, if any of the statements below hold true then the point is outside of the triangle and we can exit the function early. If all statements are false then the point is inside the triangle and we can return true.
Essentially, what is being done is determining which side the point lies in relation to the line segments making up the triangle, which is why knowing the order of the vertices is so important.

function pointInTriangle(P, A, B, C)
	v = B.x - A.x
	q = B.y - A.y
	r = C.x - B.x
	t = C.y - B.y

	rem if vertices are clockwise
	if (v*t - q*r) < 0
		if v*(P.y-A.y) >= q*(P.x-A.x) then exitfunction 0
		if r*(P.y-B.y) >= t*(P.x-B.x) then exitfunction 0
		if (A.x-C.x)*(P.y-C.y) >= (A.y-C.y)*(P.x-C.x) then exitfunction 0
	rem vertices are counter-clockwise
	else
		if v*(P.y-A.y) < q*(P.x-A.x) then exitfunction 0
		if r*(P.y-B.y) < t*(P.x-B.x) then exitfunction 0
		if (A.x-C.x)*(P.y-C.y) < (A.y-C.y)*(P.x-C.x) then exitfunction 0
	endif
endfunction 1

Distances

Misc Formulas

Addition of 2 vectors: (C = A + B)

Subtraction of 2 vectors: (C = A - B)

Dot Product of 2 vectors: (d = A dot B)

Cross Product of 2 vectors: (C = A * B)

Projection of 3D into 2D:

Reflection:

Intersection of 2 ellipses: <Example>

Intersection of 2 line segments:

Intersection of a circle and a line segment:

Intersection of an ellipse and a line segment:

Point in a capsule:

Point in a 2D triangle:

Length of a line:

Distance of a point from a line:

Distance of a point from a plane:

Pythagoras:

Projectile Motion:

Position in front of camera: