Given:
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Ball's current point: x1, y1, and z1
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A point the Ball is moving toward: x2, y2, and z2
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And speed of Ball in units/frame: n
Frame is define as the time increment between updates.
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Ball's radius: br
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Sphere's center point: sx1, sy1, and sz1
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A point the Sphere is moving toward: sx2, sy2, and sz
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Sphere's radius: sr
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And speed of Sphere in units/frame: ns
Asked to Write Code for following:
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Will the projectile hit the sphere?
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If so, where in xyz space will the collision take place?
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At what angle will the projectile ricochet off the sphere?
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Give the new velocity vector of ball.
Assumptions:
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Ball and Sphere move in stright lines.
But code could be modify for paths that follow mathematical paths defined as a function of time or space, I belive.
Like the Sphere traveling in a circle.
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Ball's current point is its starting point. It is assumed before this point it doesn't matter.
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Ball can't move backwards along its path unless it hits something maybe.
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Ball can't start inside sphere. Wouldn't make sense to test for hit when inside.
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Ball can't rotate about it center point, needed to simplify math for Ricochet angle.
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Ricochet angle (in radians) is to the tangent plain's surface
at hit point, not to the normal of the plain.
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Ball has no mass, so momentum is not factored in.
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Frame or how many frames (time) to hit, or distance to hit point was not wanted but I provide.
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Structure for the answer was not set so had to create one.
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With use of floats there is some error due to imprecision
in values of floats. See Considerations
Steps to Solve:
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Convert to Bullet and Moving Sphere Solution:
(see Bullet & Moving Sphere for explaination.)
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Find unit vector of ball and sphere
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Get line equations for X, Y, & Z using
ds as distance along line from point 1.
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Get distance equation from any point on
line to center of sphere.
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Solve for D and take derivative with respect
to ds.
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Solve for ds by setting D/ds equal to zero.
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Test if ds >= 0 then continue, if ds is neg.
means ball has to go backwards to hit sphere.
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Put ds back into distance equations and
get smallest distance form line to sphere.
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Test if D <= R then continue Hit happen,
were D is distance squared & R is radius squared.
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Use distance equation again with D = R and
solve for ds, ds will have two solutions.
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The correct ds is the smallest non negative
answer.
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Put this ds into line equations for X,
Y, & Z to get point were ball hit the sphere.
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Find unit vector form point on sphere to
center this defines the tangent vector.
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Use dot product with this line's unit vector
to find angle between ball's vector.
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Take 90 - angle found to get ricochet
angle.
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Calculate the two vectors normal to and
parallel to tangent plain on sphere though hit point.
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Reverse normal vector and add to parallel
vector to get ne ricochet vector of ball.
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Multiply by speed of ball to get new
velocity vector.
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Done.
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Good points about this method
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Considerations
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CODE
Some Notes:
This is solved by taking the ball radius and added to the sphere's and then treating the ball as
a bullet and slove for hit that way. Then Convert back to Ball to get other data.
Convert to Bullet and Moving Sphere Solution
Take ball radius and add to Sphere's.
r = sr + br
Treat Ball as point bullet.
Ball's & Sphere's unit
vectors.
Ball's unit vector is used to define the ball's path at any point
along it.
U = Ux + Uy + Uz
Find distances form point 1 to point 2 in X, Y, & Z directions.
X = x2 - x1
Y = y2 - y1
Z = z2 - z1
Find distance between to points.
D = (X^2 + Y^2 + Z^2)^.5
Find Unit Vector by dividing each component by D.
Ux = X/D
Uy = Y/D
Uz = Z/D
Sphere's unit vector is used to define the ball's path at any point along it.
S = Sx + Sy + Sz
Find distances form point 1 to point 2 in X, Y, & Z directions.
X = sx2 - sx1
Y = sy2 - sy1
Z = sz2 - sz1
Find distance between to points.
D = (X^2 + Y^2 + Z^2)^.5
Find Unit Vector by dividing each component by D.
Sx = X/D
Sy = Y/D
Sz = Z/D
Get line equations
for X, Y, & Z using ds as distance along line from point 1.
Ball's:
Xt = x1 + Ux*ds
Yt = y1 + Uy*ds
Zt = z1 + Uz*ds
Sphere's:
SXt = sx1 + Sx*ds2
SYt = sy1 + Sy*ds2
SZt = sz1 + Sz*ds2
Get distance
equation from any point on ball's line to center of sphere.
Distance equation is D = (X^2 + Y^2 + Z^2)^.5, but it can be kept in
the form of D^2 = X^2 + Y^2 + Z^2
because we can compare to R^2 which is radius of sphere squared.
X = SXt - Xt = (sx1 + Sx*ds2) - (x1 + Ux*ds)
Y = SYt - Yt = (sy1 + Sy*ds2) - (y1 + Uy*ds)
Z = SZt - Zt = (sz1 + Sz*ds2) - (z1 + Uz*ds)
D^2 = (sx1 + Sx*ds2 - x1 - Ux*ds)^2 + (sy + Sy*ds2 - y1 - Uy*ds)^2 + (sz + Sz*ds2 - z1 - Uz*ds)^2
We know the speed of both the ball and sphere. So we can find out how much ball moves compare to sphere as a single ds.
ds = n*t and ds2 = ns*t; t = ds/n = ds2/ns so for any given time interval ds2 = ds*(ns/n).
We do this to make math a lot easiler by only solving with one varible, use N=ns/n so ds2 = N*ds.
Expand out squares:
For X:
(sx1 + Sx*N*ds - x1 - Ux*ds)^2 = sx1^2 + 2*sx1(Sx*N*ds) - 2*sx1*x1 - 2*sx1(Ux*ds) + (Sx*N*ds)^2 - 2*x1(Sx*N*ds*) - 2*Ux*ds(Sx*N*ds) + x1^2 + 2*x1(Ux*ds) + (Ux*ds)^2
(sx1 + Sx*N*ds - x1 - Ux*ds)^2 = sx1^2 - 2*sx1*x1 + x1^2 + 2*sx1(Sx*N*ds) - 2*x1(Sx*N*ds*) - 2*sx1(Ux*ds) + 2*x1(Ux*ds) + (Sx*N*ds)^2 - 2*Ux*ds(Sx*N*ds) + (Ux*ds)^2
let C1 = sx1^2 - 2*sx1*x1 + x1^2
and C2 = sx1*Ux - x1*Ux
and C7 = sx1*Sx*N - x1*Sx*N
(sx1 + Sx*ds*N - x1 - Ux*ds)^2 = C1 - 2(C2 - C7)ds + (Sx^2*N^2 -2(Sx*Ux*N) + Ux^2)ds^2
For Y:
(sy - y1 - Uy*ds)^2 = sy1^2 + 2*sy1(Sy*N*ds) - 2*sy1*y1 - 2*sy1(Uy*ds) + (Sy*N*ds)^2 - 2*y1(Sy*N*ds*) - 2*Uy*ds(Sy*N*ds) + y1^2 + 2*y1(Uy*ds) + (Uy*ds)^2
(sy1 + Sy*N*ds - y1 - Uy*ds)^2 = sy1^2 - 2*sy1*y1 + y1^2 + 2*sy1(Sy*N*ds) - 2*y1(Sy*N*ds*) - 2*sy1(Uy*ds) + 2*y1(Uy*ds) + (Sy*N*ds)^2 - 2*Uy*ds(Sy*N*ds) + (Uy*ds)^2
let C3 = sy^2 - 2sy*y1 + y1^2
and C4 = sy1*Uy - y1*Uy
and C8 = sy1*Sy*N - y1*Sy*N
(sy1 + Sy*N*ds - y1 - Uy*ds)^2 = C3 - 2(C4 - C8)ds + (Sz^2*N^2 - 2(Sy*Uy*N) + Uy^2)ds^2
For Z:
(sz1 + Sz*N*ds - z1 - Uz*ds)^2 = sz1^2 + 2*sz1(Sz*N*ds) - 2*sz1*z1 - 2*sz1(Uz*ds) + (Sz*N*ds)^2 - 2*z1(Sz*N*ds*) - 2*Uz*ds(Sz*N*ds) + z1^2 + 2*z1(Uz*ds) + (Uz*ds)^2
(sz1 + Sz*N*ds - z1 - Uz*ds)^2 = sz1^2 - 2*sz1*z1 + z1^2 + 2*sz1(Sz*N*ds) - 2*z1(Sz*N*ds*) - 2*sz1(Uz*ds) + 2*z1(Uz*ds) + (Sz*N*ds)^2 - 2*Uz*ds(Sz*N*ds) + (Uz*ds)^2
let C5 = sz1^2 - 2sz*z1 + z1^2
and C6 = sz1*Uz - z1*Uz
and C9 = sz1*Sz*N - z1*Sz*N
(sz1 + Sz*N*ds - z1 - Uz*ds)^2 = C5 - 2(C6 - C9)ds + (Sz^2*N^2 - 2(Sz*Uz*N) + Uz^2)ds^2
Note:
Ux^2 + Uy^2 + Uz^2 = 1 because U is a unit vector.
Sx^2 + Sy^2 + Sz^2 = 1 because S is also a unit vector
So the ds^2 terms can be rewritten to:
(Sx^2 + Sy^2 + Sz^2)N^2 -2(Sx*Ux + Sy*Uy + Sz*Uz)N + (Ux^2 + Uy^2 + Uz^2)
and simplified to:
let C10 = N - 2*N(Sx*Ux + Sy*Uy + Sz*Uz) + 1
So final D^2 is:
D^2 = C1 + C3 + C5 - 2(C2 + C4 + C6 - (C7 + C8 + C9))*ds + C10*ds^2
Solve for
D and take derivative with respect to ds.
D = (C1 + C3 + C5 - 2(C2 + C4 + C6 - (C7 + C8 + C9))*ds + C10*ds^2)^.5
Let C11 = C1 + C3 + C5 and C12 = C2 + C4 + C6 - (C7 + C8 + C9)
dD = (.5(C11 - 2(C12)*ds + C10*ds^2)^-.5)
*(-2(C12) + 2*C10*ds)
dD = ((-C12 + C10*ds) / ((C11 - 2(C12)*ds + C10*ds^2)^.5)
Solve for
ds by setting dD equal to zero.
We set dD to zero because that is were the distance form line to Sphere's
center is smallest
and the only way dD can be zero is if -C12 + C10*ds = 0
So ds = C12 / C10
Test if ds
>= 0 then continue, if ds is neg. means ball has to go backwards to hit
sphere.
ds could be negative which would mean the line is closest behind point
1.
Since the ball starts at point 1 and moves to pt 2 there can be no
hit.
Put ds back
into distance equations and get smallest distance form line to sphere.
Put ds back into Distance square: D2 = C11 - 2(C12)*ds + C10*ds^2
D2 = C11 - 2(C12)*(C12/C10) + C10*(C12/C10)*(C12/C10)
D2 = C11 - C12^2/C10
Test if D
<= R then continue Hit happen, where D is distance squared & R is
radius squared.
So if D2 which is distance squared (D^2) form center to closest point
on line is less than R2 = r^2 then we must have hit the Sphere.
Use distance
equation again with D = R and solve for ds, ds will have two solutions.
Now set D2 the value of R2 and solve for the two ds's that will solve
the equitation.
Use the aX^2 + bX + c = 0 solution of
X = (-b - (b^2 + 4ac)^.5)/2a
X = (-b + (b^2 + 4ac)^.5)/2a
Set a variable sq to (b^2 + 4ac)^.5
R2 = C11 - 2(C12)*ds + C10*ds^2
C10*ds^2 -2(C12)ds + C11 - R2; a=C10, b=-2*(C12), and c = C11 - R2
sq = (4(C12)*(C12) + 4*C10*(C11 - R2))^.5
Solve or two ds's
ds = (-2*(C12) - sq )/(2*C10)
or
ds = (-2*(C12) + sq)/(2*C10)
The correct
ds is the smallest non negative answer.
This is for a check just in case point one might have been in Sphere
to begin with.
Put this
ds into line equations for X, Y, & Z to get point were ball hit the sphere.
Xt = x1 + Ux*ds
Yt = y1 + Uy*ds
Zt = z1 + Uz*ds
So hit point happens at Ball's Center (Xt, Yt, Zt)
time to hit is time = ds/n and frames till hit is frames = time*fps
Now find Sphere's center at hit
SXt = sx + Sx*N*ds
SYt = sy + Sy*N*ds
SZt = sz + Sz*N*ds
Find unit
vector form point on sphere to center.
Now we need to find the angle the ball has to the tangent plain of
the sphere at the hit point.
First we need the normal to this tangent plain which is the unit vector
form sphere's center to hit point.
T = Tx + Ty + Tz
X = SXt - Xt
Y = SXt - Yt
Z = SXt - Zt
D = (X^2 + Y^2 + Z^2)^.5
Tx = X/D
Ty = Y/D
Tz = Z/D
Convert back to ball and moving sphere
We have the hit spot of where a ball hit the Sphere of radius r = rs + br
The hit spot is where the center of ball now a point hit the sphere.
We now have to convert back to Ball hitting Sphere.
Start at Ball's center point and add the ball's radius times Unit vector of line to get point on Sphere surface were hit oocurs.
X = Xt + br*Tx
Y = Yt + br*Ty
Z = Zt + br*Tz
So hit point where Ball and Sphere surface hit is (X, Y, Z)
Use Dot
product with lines unit vector to find angle between lines vector and sphere
vector.
Now that we have unit vector of tangent plain we can get the angle
between the ball's unit vector and plains.
This is done by use of the Dot product.
cos(theta) = (U dot T)/(|U||T|)
since U and T are unit vectors |U| = 1 and |T| = 1
So cos(theta) = U dot V solve for theta
and U dot T = Ux*Tx + Uy*Ty + Uz*Tz
theta = acos(Ux*Tx + Uy*Ty + Uz*Tz) might be different depending on
complier.
Take 90
- angle found to get ricochet angle.
The theta found is the angle from ball's path to the normal to tangent
plain.
The ricochet angle is pi_half - theta.
theta is given in terms of radians and pi_half = 1.570796326795.
ricochet angle = pi_half - theta in radians.
Calculate
the two vectors normal to and parallel to tangent plain on sphere though
hit point.
Now we need the two components perpendicular and parallel to the tangent
plain.
Uprep = U sin(ricochet angle)
Upara = U - Uperp
Reverse
normal vector and add to parallel vector to get new ricochet vector of
ball.
ball's Upara will stay the same but its Uperp will point in opposite
direction which means we take negative of it.
Then add the two unit vector back together to get new unit vector in
direction of ricochet.
U = Upara - Uperp
Multiply
by speed of ball to get new velocity vector.
Take this new U and multiply by n the speed of ball in frames to
get new velocity vector.
Done
You now know if the Ball hit the Moving Sphere.
Were in 3D space it hit the Sphere.
The angle of ricochet of the Ball to tangent plain surface
and the velocity vector of Ball's new path.
Good
points about this method
This is a good way to test for a hit because you don't need to check
each frame for a hit and you
can easily find out what frame the hit occurs by take distance form
point 1 to sphere (ds) divid by speed
of ball (n). and added to your current frame.
Considerations
Some things to consider about this are:
With the use of floats there is some error due to floats not being
precise ie( 0 could be 1.453e-23).
Hit could happen between frames.
This could be change for cylinder and/or box test also.
CODE
Code uses:
3 square root calls
1 acos (arccos) call
1 sin call
Every thing else is adds, subtractions, multiplies and divides.
At every possible chance test for possible hit is done and if not possible
return FALSE.
Code is in C and all variables are global for speed with two structs
created for 3d points and the answer. |