class Geometric in Tinman.Core.Math

Helper class for computing geometric intersections.

static class Geometric

Public / Methods

AngleAdd

Add the given cosine angles.

`[Pure]`
`public static method`	`AngleAdd (float64 a, float64 b)`
`type`	`float64`
`params`	`a`		Cosine of first angle to add.
	`b`		Cosine of second angle to add.
`returns`		`[-1..1]`	Cosine of added angle.

Remarks:

This method computes cos(acos(a) + acos(b)) without trigonometric functions: a * b - sqrt(1 - a^2) * sqrt(1 - b^2)

AngleFlip

Flips the given angle at 90°.

`[Pure]`
`public static method`	`AngleFlip (float64 a)`
`type`	`float64`
`params`	`a`		Cosine of angle to flip.
`returns`		`[0..1]`	Cosine of flipped angle.

Remarks:

This method computes cos(90° - acos(a)) without trigonometric functions: sqrt(1 - a^2)

AngleSub

Subtracts the given cosine angles.

`[Pure]`
`public static method`	`AngleSub (float64 a, float64 b)`
`type`	`float64`
`params`	`a`		Cosine of first angle to subtract in the range.
	`b`		Cosine of second angle to subtract.
`returns`		`[-1..1]`	Cosine of subtracted angle.

Remarks:

This method computes cos(acos(a) - acos(b)) without trigonometric functions: a * b + sqrt(1 - a^2) * sqrt(1 - b^2)

For common values of a, the formula reduces to these forms:

cos(90°) => sqrt(1 - b^2)
cos(180°) => -b

Area

Computes the area of the given triangle.

`[Pure]`
`public static method`	`Area (Vec2D v1, Vec2D v2, Vec2D v3)`
`type`	`float64`
`params`	`v1`	First vertex of triangle.
	`v2`	Second vertex of triangle.
	`v3`	Third vertex of triangle.
`returns`		The signed area of the triangle.

Computes the area of the given triangle.

`[Pure]`
`public static method`	`Area (Vec3D v1, Vec3D v2, Vec3D v3)`
`type`	`float64`
`params`	`v1`		First vertex of triangle.
	`v2`		Second vertex of triangle.
	`v3`		Third vertex of triangle.
`returns`		`[>=0]`	The area of the triangle.

Computes the signed area of the given simple polygon.

`public static method`	`Area (Vec2D[] v, int32 a, int32 b)`
`type`	`float64`
`params`	`v`	`[not-null]`	The polygon vertices.
	`a`	`[0..v.Length-1]`	Index of first polygon vertex.
	`b`	`[0..v.Length-1]`	Index of last polygon vertex.
`returns`			The signed area of the simple polygon.

Remarks:

The polygon is defined as follows:

v[a    ] := first polygon vertex
v[a + n] := n-th polygon vertex
v[b    ] := last polygon vertex (not equal to first)

where b - a >= 2
      b - a >= n >= 0

Inverting the order of the polygon vertices will flip the sign of the area value. For a standard Cartesian coordinate system (X-axis points to the right, Y-axis points upwards), clockwise polygons have negative area values and counter-clockwise polygons have positive area values.

Computes the area of the given triangle.

`[Pure]`
`public static method`	`Area (float64 x1, float64 y1, float64 x2, float64 y2, float64 x3, float64 y3)`
`type`	`float64`
`params`	`x1`	X-coordinate of first vertex of triangle.
	`y1`	Y-coordinate of first vertex of triangle.
	`x2`	X-coordinate of second vertex of triangle.
	`y2`	Y-coordinate of second vertex of triangle.
	`x3`	X-coordinate of third vertex of triangle.
	`y3`	Y-coordinate of third vertex of triangle.
`returns`		The signed area of the triangle.

Computes the area of the given triangle.

`[Pure]`
`public static method`	`Area (float64 x1, float64 y1, float64 z1, float64 x2, float64 y2, float64 z2, float64 x3, float64 y3, float64 z3)`
`type`	`float64`
`params`	`x1`		X-coordinate of first vertex of triangle.
	`y1`		Y-coordinate of first vertex of triangle.
	`z1`		Z-coordinate of first vertex of triangle.
	`x2`		X-coordinate of second vertex of triangle.
	`y2`		Y-coordinate of second vertex of triangle.
	`z2`		Z-coordinate of second vertex of triangle.
	`x3`		X-coordinate of third vertex of triangle.
	`y3`		Y-coordinate of third vertex of triangle.
	`z3`		Z-coordinate of third vertex of triangle.
`returns`		`[>=0]`	The area of the triangle.

CartesianToSpherical

Converts the given cartesian coordinates into spherical coordinates.

`[Pure]`
`public static method`	`CartesianToSpherical (Vec3D cartesian)`
`type`	`Vec3D`
`params`	`cartesian`	The cartesian coordinates.
`returns`		The spherical coordinates.

Remarks:

The components of the returned vector have the following semantic:

spherical.X := radius (distance to origin).
spherical.Y := azimuthal angle (angle from positive X-axis in radians) in the range [-PI..PI].
spherical.Z := polar angle (angle from positive Z-axis in radians) in the range [0..PI].

Converts the given cartesian coordinates into spherical coordinates.

`[Pure]`
`public static method`	`CartesianToSpherical (float64 x, float64 y, float64 z)`
`type`	`Vec3D`
`params`	`x`	The cartesian X-coordinate.
	`y`	The cartesian Y-coordinate.
	`z`	The cartesian Z-coordinate.
`returns`		The spherical coordinates.

Remarks:

The components of the returned vector have the following semantic:

spherical.X := radius (distance to origin).
spherical.Y := azimuthal angle (angle from positive X-axis in radians) in the range [-PI..PI].
spherical.Z := polar angle (angle from positive Z-axis in radians) in the range [0..PI].

CheckIntersectionOfLineSegments

Checks if the given line segments intersect.

`[Pure]`
`public static method`	`CheckIntersectionOfLineSegments (Vec2D v1, Vec2D v2, Vec2D v3, Vec2D v4)`
`type`	`bool`
`params`	`v1`	Start of first line segment.
	`v2`	End of first line segment.
	`v3`	Start of second line segment.
	`v4`	End of second line segment.
`returns`		`true` if the line segments intersect, `false` if not.

Checks if the given line segments intersect.

`[Pure]`
`public static method`	`CheckIntersectionOfLineSegments (float64 x1, float64 y1, float64 x2, float64 y2, float64 x3, float64 y3, float64 x4, float64 y4)`
`type`	`bool`
`params`	`x1`	X-coordinate of start of first line segment.
	`y1`	Y-coordinate of start of first line segment.
	`x2`	X-coordinate of end of first line segment.
	`y2`	Y-coordinate of end of first line segment.
	`x3`	X-coordinate of start of second line segment.
	`y3`	X-coordinate of start of second line segment.
	`x4`	X-coordinate of end of second line segment.
	`y4`	X-coordinate of end of second line segment.
`returns`		`true` if the line segments intersect, `false` if not.

DistanceLineToLine

Computes the pair of nearest points on two lines.

`[Pure]`
`public static method`	`DistanceLineToLine (Vec3D s1, Vec3D v1, Vec3D s2, Vec3D v2)`
`type`	`Vec2D`
`params`	`s1`	Point on first line.
	`v1`	Direction vector of first line.
	`s2`	Point on second line.
	`v2`	Direction vector of second line.
`returns`		The coefficient vector for computing the two nearest points on both lines: p1 = s1 + v1 * X p2 = s2 + v2 * Y

DistancePointToHorizon

Computes the distance from the given point to the horizon circle on a sphere centered at the coordinate origin and having the given radius.

`[Pure]`
`public static method`	`DistancePointToHorizon (Vec3D point, float64 radius)`
`type`	`float64`
`params`	`point`		The point coordinates.
	`radius`	`[>0]`	The sphere radius.
`returns`		`[>=0]`	The distance to the horizon, in terrain-space.