// projectile in principal space
// Create a principalSpace3D and PrincipalTrajectory from a given Projectile3D,
// the launch position and another point on the plane (e.g. a 3d target).
// Implements Equation 10 from the paper.
public static PrincipalSpace3D Create(
Projectile3D projectile3D, // projectile in world space
Vector3 worldP0, // the launch position
Vector3 worldP1, // any other position on the principal plane
out PrincipalProjectile principalProjectile)
{
principalProjectile = new PrincipalProjectile(
projectile3D.k, projectile3D.vInfinity.magnitude);
Vector3 yAxis = projectile3D.vInfinity / -principalProjectile.vInfinity;
Vector3 deltaPos = worldP1 - worldP0;
Vector3 xAxis = deltaPos - Vector3.Dot(yAxis, deltaPos) * yAxis;
float squaredXScale = xAxis.sqrMagnitude;
if (squaredXScale == 0)
{
xAxis = Misc.GetAnyVector3PerpendicularTo(yAxis);
squaredXScale = xAxis.sqrMagnitude;
}
xAxis /= Mathf.Sqrt(squaredXScale);
return new PrincipalSpace3D(worldP0, xAxis, yAxis);
}
// Create a principalSpace3D and PrincipalTrajectory from a given Projectile3D,
// the launch position and another point on the plane (e.g. a 3d target).
// Implements Equation 10 from the paper.
public static PrincipalSpace3D Create(
Projectile3D projectile3D, // projectile in world space
Vector3 worldP0, // the launch position
Vector3 worldP1, // any other position on the principal plane
out PrincipalProjectile principalProjectile) // projectile in principal space
{
principalProjectile = new PrincipalProjectile(
projectile3D.k, projectile3D.vInfinity.magnitude);
Vector3 yAxis = projectile3D.vInfinity / -principalProjectile.vInfinity;
Vector3 deltaPos = worldP1 - worldP0;
Vector3 xAxis = deltaPos - Vector3.Dot(yAxis, deltaPos) * yAxis;
float squaredXScale = xAxis.sqrMagnitude;
if (squaredXScale == 0)
{
xAxis = Misc.GetAnyVector3PerpendicularTo(yAxis);
squaredXScale = xAxis.sqrMagnitude;
}
xAxis /= Mathf.Sqrt(squaredXScale);
return new PrincipalSpace3D(worldP0, xAxis, yAxis);
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory that also hits the line exactly 'b' above the
// line from the origin to r. This is equivalent to saying that the trajectory
// will touch the line y(x) = r.y/r.x * x + b, and that it is exactly contained
// by the parallelogram formed by the vectors (0, 0)-(r.x, r.y) and (0, 0)-(0, b)
// Uses Equation 27 from the paper.
public static float GetTimeToTargetRGivenArcHeightB(
PrincipalProjectile projectile, Vector2 r, float b)
{
return(GetTimeToTargetRGivenArcHeightB(
projectile.k, projectile.vInfinity, r.x, r.y, b));
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory that also passes through position q.
// Uses Equation 23 from the paper.
public static float GetTimeToTargetRGivenIntermediatePositionQ(
PrincipalProjectile projectile, Vector2 r, Vector2 q)
{
return(GetTimeToTargetRGivenIntermediatePositionQ(
projectile.k, projectile.vInfinity, r.x, r.y, q.x, q.y));
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory with the initial velocity s. When s is too small
// to hit r at all, the functions return 0. But when it great enough, the
// problem has two solutions: One arc that's higher and one arc that's lower
// than the one that's found by GetTimeToTargetRWithMinimalInitialSpeed().
// Use highArc to select which one to return the time for.
// Uses Equation 30 from the paper.
public static float GetTimeToTargetRGivenInitialSpeedS(
PrincipalProjectile projectile, Vector2 r, double s, bool highArc)
{
return((float)GetTimeToTargetRGivenInitialSpeedS(
projectile.k, projectile.vInfinity, r.x, r.y, s, highArc));
}
public PrincipalTrajectory(PrincipalProjectile principalProjectile, Vector2 v0) :
base(principalProjectile)
{
this.v0 = v0;
}
public PrincipalTrajectory(PrincipalProjectile principalProjectile, Vector2 v0)
: base(principalProjectile)
{
this.v0 = v0;
}
public float vInfinity; // velocity of projectile falling indefinitely
public PrincipalProjectile(PrincipalProjectile principalProjectile)
{
k = principalProjectile.k;
vInfinity = principalProjectile.vInfinity;
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory that has slope a at position r. Use
// principalSpace3D.ToPrincipalSlope() to get the principal slope a from
// a slope in world space. Uses Equation 26 from the paper.
public static float GetTimeToTargetRGivenTargetSlopeA(
PrincipalProjectile projectile, Vector2 r, float a)
{
return(GetTimeToTargetRGivenTargetSlopeA(
projectile.k, projectile.vInfinity, r.x, r.y, a));
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory that also passes through position q.
// Uses Equation 23 from the paper.
public static float GetTimeToTargetRGivenIntermediatePositionQ(
PrincipalProjectile projectile, Vector2 r, Vector2 q)
{
return GetTimeToTargetRGivenIntermediatePositionQ(
projectile.k, projectile.vInfinity, r.x, r.y, q.x, q.y);
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory with the initial velocity s. When s is too small
// to hit r at all, the functions return 0. But when it great enough, the
// problem has two solutions: One arc that's higher and one arc that's lower
// than the one that's found by GetTimeToTargetRWithMinimalInitialSpeed().
// Use highArc to select which one to return the time for.
// Uses Equation 30 from the paper.
public static float GetTimeToTargetRGivenInitialSpeedS(
PrincipalProjectile projectile, Vector2 r, double s, bool highArc)
{
return (float)GetTimeToTargetRGivenInitialSpeedS(
projectile.k, projectile.vInfinity, r.x, r.y, s, highArc);
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory with curviness 'h'. h is defined as the ratio between
// the height of the parallelogram (as defined in
// GetTimeToTargetRGivenArcHeightB()) and the distance to the target.
// Consequently, small values result in low arcs, and high values result in high
// arcs. h = 0.25 roughly approximates a 'minimal effort' arc.
// Uses Equation 27 and 28 from the paper.
public static float GetTimeToTargetRGivenCurvinessH(
PrincipalProjectile projectile, Vector2 r, float h)
{
return GetTimeToTargetRGivenCurvinessH(
projectile.k, projectile.vInfinity, r.x, r.y, h);
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory that also hits the line exactly 'b' above the
// line from the origin to r. This is equivalent to saying that the trajectory
// will touch the line y(x) = r.y/r.x * x + b, and that it is exactly contained
// by the parallelogram formed by the vectors (0, 0)-(r.x, r.y) and (0, 0)-(0, b)
// Uses Equation 27 from the paper.
public static float GetTimeToTargetRGivenArcHeightB(
PrincipalProjectile projectile, Vector2 r, float b)
{
return GetTimeToTargetRGivenArcHeightB(
projectile.k, projectile.vInfinity, r.x, r.y, b);
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory with the smallest possible initial velocity.
// Uses Equation 29 from the paper.
public static float GetTimeToTargetRWithMinimalInitialSpeed(
PrincipalProjectile projectile, Vector2 r)
{
return (float)GetTimeToTargetRWithMinimalInitialSpeed(
projectile.k, projectile.vInfinity, r.x, r.y);
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory with curviness 'h'. h is defined as the ratio between
// the height of the parallelogram (as defined in
// GetTimeToTargetRGivenArcHeightB()) and the distance to the target.
// Consequently, small values result in low arcs, and high values result in high
// arcs. h = 0.25 roughly approximates a 'minimal effort' arc.
// Uses Equation 27 and 28 from the paper.
public static float GetTimeToTargetRGivenCurvinessH(
PrincipalProjectile projectile, Vector2 r, float h)
{
return(GetTimeToTargetRGivenCurvinessH(
projectile.k, projectile.vInfinity, r.x, r.y, h));
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory that touches the line y(x) = a*x + b. Use
// principalSpace3D.ToPrincipalIntersectionLine() to get the (a,b) line
// parameters from a world space plane. Uses Equation 24 from the paper.
public static float GetTimeToTargetRGivenLineToTouch(
PrincipalProjectile projectile, Vector2 r, Vector2 lineAB)
{
return GetTimeToTargetRGivenLineToTouch(
projectile.k, projectile.vInfinity, r.x, r.y, lineAB.x, lineAB.y);
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory that touches the line y(x) = a*x + b. Use
// principalSpace3D.ToPrincipalIntersectionLine() to get the (a,b) line
// parameters from a world space plane. Uses Equation 24 from the paper.
public static float GetTimeToTargetRGivenLineToTouch(
PrincipalProjectile projectile, Vector2 r, Vector2 lineAB)
{
return(GetTimeToTargetRGivenLineToTouch(
projectile.k, projectile.vInfinity, r.x, r.y, lineAB.x, lineAB.y));
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory that has slope a at position r. Use
// principalSpace3D.ToPrincipalSlope() to get the principal slope a from
// a slope in world space. Uses Equation 26 from the paper.
public static float GetTimeToTargetRGivenTargetSlopeA(
PrincipalProjectile projectile, Vector2 r, float a)
{
return GetTimeToTargetRGivenTargetSlopeA(
projectile.k, projectile.vInfinity, r.x, r.y, a);
}
// Get the time at which the (principal-space) target position r will be hit
// as part of a trajectory with the smallest possible initial velocity.
// Uses Equation 29 from the paper.
public static float GetTimeToTargetRWithMinimalInitialSpeed(
PrincipalProjectile projectile, Vector2 r)
{
return((float)GetTimeToTargetRWithMinimalInitialSpeed(
projectile.k, projectile.vInfinity, r.x, r.y));
}
public Vector2 v0; // projectile velocity at t = 0
public PrincipalTrajectory(PrincipalProjectile principalProjectile) :
base(principalProjectile)
{
}
public float vInfinity; // velocity of projectile falling indefinitely
#endregion Fields
#region Constructors
public PrincipalProjectile(PrincipalProjectile principalProjectile)
{
k = principalProjectile.k;
vInfinity = principalProjectile.vInfinity;
}
public Vector2 v0; // projectile velocity at t = 0
#endregion Fields
#region Constructors
public PrincipalTrajectory(PrincipalProjectile principalProjectile)
: base(principalProjectile)
{
}
