I am implementing a very simple animation effect for a game. The scenario is like this:
there is a elastic rubber line, length is 1 meter, when it is extended over 1 meter, it is elastic.
the line connects two dots A and B like this, the distance is S, S > 1 meter
A <------------- B
then fix dot A, and releases B, the line takes B to the direction of A
I want to know how to calculate time T, which B costs to move X meters towards A (X <= S).
Any ideas?
Thanks!
I have been meaning to learn how to animate these kinds of images in sage (a python based platform for math) for a while, so i used this as an excuse. I hope this code snippet and image is helpful.
A = 3
w = 0.5
# x = f(t) = A cos(wt) inside elastic region
# with x = displacement from 1 meter mark
# in the below code, x is the displacement from origin (x = A cos(wt) + 1)
# find speed when we cross the one meter mark
# f'(t) = -Aw sin(wt), but this is also max speed
# ie f'(t at one meter mark) = -Aw
speed_max = -A * w
# time to reach max speed + time to cross last meter
eta = float(pi/2 * 1/w + 1/abs(speed_max))
# the function you were looking for
def time_left(x):
if x < 1:
return x/abs(speed_max)
else:
return 1/w * arccos((x-1)/A)
It may not be clear in the image but within one meter of the origin there is no acceleration.
Related
I have been trying to work out how to calculate Poincaré sections for a system of non-linear ODEs, using a paper on the exact system as reference, and have been wrestling with numpy to try and make it run better. This is intended to run within a bounded domain.
Currently, I have the following code
import numpy as np
from scipy.integrate import odeint
X = 0
Y = 1
Z = 2
def generate_poincare_map(function, initial, plane, iterations, delta):
intersections = []
p_i = odeint(function, initial.flatten(), [0, delta])[-1]
for i in range(1, iterations):
p_f = odeint(function, p_i, [i * delta, (i+1) * delta])[-1]
if (p_f[Z] > plane) and (p_i[Z] < plane):
intersections.append(p_i[:2])
if (p_f[Z] > plane) and (p_i[Z] < plane):
intersections.append(p_i[:2])
p_i = p_f
return np.stack(intersections)
This is pretty wasteful due to the integration solely between successive time steps, and seems to produce incorrect results. The original reference includes sections along the lines of
whereas mine tend to result in something along the lines of
Do you have any advice on how to proceed to make this more correct, and perhaps a little faster?
To get a Pointcaré map of the ABC flow
def ABC_ode(u,t):
A, B, C = 0.75, 1, 1 # matlab parameters
x, y, z = u
return np.array([
A*np.sin(z)+C*np.cos(y),
B*np.sin(x)+A*np.cos(z),
C*np.sin(y)+B*np.cos(x)
])
def mysolver(u0, tspan): return odeint(ABC_ode, u0, tspan, atol=1e-10, rtol=1e-11)
you have first to understand that the dynamical system is really about the points (cos(x),sin(x)) etc. on the unit circle. So values different by multiples of 2*pi represent the same point. In the computation of the section one has to reflect this, either by computing it on the Cartesian product of the 3 circles. Let's stay with the second variant, and chose [-pi,pi] as the fundamental period to have the zero location well in the center. Keep in mind that jumps larger pi are from the angle reduction, not from a real crossing of that interval.
def find_crosssections(x0,y0):
u0 = [x0,y0,0]
px = []
py = []
u = mysolver(u0, np.arange(0, 4000, 0.5)); u0 = u[-1]
u = np.mod(u+pi,2*pi)-pi
x,y,z = u.T
for k in range(len(z)-1):
if z[k]<=0 and z[k+1]>=0 and z[k+1]-z[k]<pi:
# find a more exact intersection location by linear interpolation
s = -z[k]/(z[k+1]-z[k]) # 0 = z[k] + s*(z[k+1]-z[k])
rx, ry = (1-s)*x[k]+s*x[k+1], (1-s)*y[k]+s*y[k+1]
px.append(rx);
py.append(ry);
return px,py
To get a full picture of the Poincare cross-section and avoid duplicate work, use a grid of squares and mark if one of the intersections already fell in it. Only start new iterations from the centers of free squares.
N=20
grid = np.zeros([N,N], dtype=int)
for i in range(N):
for j in range(N):
if grid[i,j]>0: continue;
x0, y0 = (2*i+1)*pi/N-pi, (2*j+1)*pi/N-pi
px, py = find_crosssections(x0,y0)
for rx,ry in zip(px,py):
m, n = int((rx+pi)*N/(2*pi)), int((ry+pi)*N/(2*pi))
grid[m,n]=1
plt.plot(px, py, '.', ms=2)
You can now play with the density of the grid and the length of the integration interval to get the plot a little more filled out, but all characteristic features are already here. But I'd recommend re-programming this in a compiled language, as the computation will take some time.
SHM - Simple Harmonic Motion
A simple harmonic oscillator has a frequency of 6.2 Hz. It is oscillating along x, where x(t) = Acos(wt + d). You are given the velocity at two moments: v(t=0) = 1.9 cm/s and v(t=.1) = -19.4 cm/s.
Calculate A and d.
Since I know that T = 1/f, I found out that T = 0.1612
And since, W = 2pi / T, w = 38.9557
Can't find out what to do with the velocities ?!
Need help as soon as possible...
You get
v(t) = -A*w*sin(w*t+d)
and you know v(0)=-Aw*sin(d) as well as v(t) for t=0.1.
v(t) = x'(t) = -A*w*( cos(w*t)*sin(d)+sin(w*t)*cos(d) )
so that
-A*w*cos(d) = ( v(t)-cos(w*t)*v(0) ) / sin(w*t)
and then conversion to polar coordinates will give you the values for A and d.
I have solid object that is spinning with a torque W, and I want to calculate the force F applied on a certain point that's D units away from the center of the object. All these values are represented in Vector3 format (x, y, z)
I know until now that W = D x F, where x is the cross product, so by expanding this I get:
Wx = Dy*Fz - Dz*Fy
Wy = Dz*Fx - Dx*Fz
Wz = Dx*Fy - Dy*Fx
So I have this equation, and I need to find (Fx, Fy, Fz), and I'm thinking of using the Simplex method to solve it.
Since the F vector can also have negative values, I split each F variable into 2 (F = G-H), so the new equation looks like this:
Wx = Dy*Gz - Dy*Hz - Dz*Gy + Dz*Hy
Wy = Dz*Gx - Dz*Hx - Dx*Gz + Dx*Hz
Wz = Dx*Gy - Dx*Hy - Dy*Gx + Dy*Hx
Next, I define the simplex table (we need <= inequalities, so I duplicate each equation and multiply it by -1.
Also, I define the objective function as: minimize (Gx - Hx + Gy - Hy + Gz - Hz).
The table looks like this:
Gx Hx Gy Hy Gz Hz <= RHS
============================================================
0 0 -Dz Dz Dy -Dy <= Wx = Gx
0 0 Dz -Dz -Dy Dy <= -Wx = Hx
Dz -Dz 0 0 Dx -Dx <= Wy = Gy
-Dz Dz 0 0 -Dx Dx <= -Wy = Hy
-Dy Dy Dx -Dx 0 0 <= Wz = Gz
Dy -Dy -Dx Dx 0 0 <= -Wz = Hz
============================================================
1 -1 1 -1 1 -1 0 = Z
The problem is that when I run it through an online solver I get Unbounded solution.
Can anyone please point me to what I'm doing wrong ?
Thanks in advance.
edit: I'm sure I messed up some signs somewhere (for example the Z should be defined as a max), but I'm sure I'm wrong when defining something more important.
There exists no unique solution to the problem as posed. You can only solve for the tangential projection of the force. This comes from the properties of the vector (cross) product - it is zero for collinear vectors and in particular for the vector product of a vector by itself. Therefore, if F is a solution of W = r x F, then F' = F + kr is also a solution for any k:
r x F' = r x (F + kr) = r x F + k (r x r) = r x F
since the r x r term is zero by the definition of vector product. Therefore, there is not a single solution but rather a whole linear space of vectors that are solutions.
If you restrict the solution to forces that have zero projection in the direction of r, then you could simply take the vector product of W and r:
W x r = (r x F) x r = -[r x (r x F)] = -[(r . F)r - (r . r)F] = |r|2F
with the first term of the expansion being zero because the projection of F onto r is zero (the dot denotes scalar (inner) product). Therefore:
F = (W x r) / |r|2
If you are also given the magnitude of F, i.e. |F|, then you can compute the radial component (if any) but there are still two possible solutions with radial components in opposing directions.
Quick dirty derivation...
Given D and F, you get W perpendicular to them. That's what a cross product does.
But you have W and D and need to find F. This is a bad assumption, but let's assume F was perpendicular to D. Call it Fp, since it's not necessarily the same as F. Ignoring magnitudes, WxD should give you the direction of Fp.
This ignoring magnitudes, so fix that with a little arithmetic. Starting with W=DxF applied to Fp:
mag(W) = mag(D)*mag(Fp) (ignoring geometry; using Fp perp to D)
mag(Fp) = mag(W)/mag(D)
Combining the cross product bit for direction with this stuff for magnitude,
Fp = WxD / mag(WxD) * mag(Fp)
Fp = WxD /mag(W) /mag(D) *mag(W) /mag(D)
= WxD / mag(D)^2.
Note that given any solution Fp to W=DxF, you can add any vector proportional to D to Fp to obtain another solution F. That is a totally free parameter to choose as you like.
Note also that if the torque applies to some sort of axle or object constrained to rotate about some axis, and F is applied to some oddball lever sticking out at a funny angle, then vector D points in some funny direction. You want to replace D with just the part perpendicular to the axle/axis, otherwise the "/mag(D)" part will be wrong.
So from your comment is clear that all rotations are spinning around center of gravity
in that case
F=M/r
F force [N]
M torque [N/m]
r scalar distance between center of rotation [m]
this way you know the scalar size of your Force
now you need the direction
it is perpendicular to rotation axis
and it is the tangent of the rotation in that point
dir=r x axis
F = F * dir / |dir|
bolds are vectors rest is scalar
x is cross product
dir is force direction
axis is rotation axis direction
now just change the direction according to rotation direction (signum of actual omega)
also depending on your coordinate system setup
so ether negate F or not
but this is in 3D free rotation very unprobable scenario
the object had to by symmetrical from mass point of view
or initial driving forces was applied in manner to achieve this
also beware that after first hit with any interaction Force this will not be true !!!
so if you want just to compute Force it generate on certain point if collision occurs is this fine
but immediately after this your spinning will change
and for non symmetric objects the spinning will be most likely off the center of gravity !!!
if your object will be disintegrated then you do not need to worry
if not then you have to apply rotation and movement dynamics
Rotation Dynamics
M=alpha*I
M torque [N/m]
alpha angular acceleration
I quadratic mass inertia for actual rotation axis [kg.m^2]
epislon''=omega'=alpha
' means derivation by time
omega angular speed
epsilon angle
Ok so let me try to explain this the best way that i can.
I have two points plotted 'A' and 'B' and I am trying to plot a third point 'C' so that it is past point 'B' but along the same slope. I have the angle of the line and I would post some code but I really have no idea where to begin.
any help would be awesome!
Just a little code that i do have
CGPoint vector = ccpSub(touchedPoint, fixedPoint);
CGFloat rotateAngle = -ccpToAngle(vector);
Assuming that by this you mean you need a 3rd point C added such that all the points are colinear, all you need to do is calculate the vector that takes you from A to B, and then generate a new point by adding multiples of this vector to the point B. Choose the multiple based on the distance you want C to be from B.
As an example, say A = (2,2), B = (4,3). Then the vector from A to B is given by (2,1).
All you need to do then is work out how far your new point is from B and add a multiple K*(2,1) to your point B where K is chosen to meet the requirements of your distance
I am assuming you are in 2D, but the same method would apply in higher dimensions
My math is rusty, but the linear equation is generally represented as y=m*x+b, where m is the slope, and b is the y-intercept. You can get m, the slope, by taking the difference of the y values and dividing that by the difference in the x values, e.g., if A = (2,2) and B = (4,3), then m is (3-2)/(4-2) or 0.5. Then, you can solve the linear equation for b, the y-intercept, i.e. b=y-m*x and then plug in either of the data points, e.g. if we plug in the x and y values for point A, you get b = 2 - 0.5 * 2 = 1. Now knowing the slope, m (0.5 in this example), and the y-intercept, b (1 in this example), you can calculate the y for any x value using y=m*x+b, in this case y=0.5*x+1.
So, if touchedPoint and fixedPoint are CGPoint, you can calculate the slope and y-intercept from fixedPoint and touchedPoint like so:
double m = (fixedPoint.y - touchedPoint.y) / (fixedPoint.x - touchedPoint.x);
double b = fixedPoint.y - m * fixedPoint.x;
Now, you don't say how you want to determine where this third point, C, is. But if you, for example, knew the x coordinate for this new point C, you can calculate the y coordinate that falls on the same line as follows:
CGPoint pointC;
pointC.x = 400; // or set this to whatever you want
pointC.y = m * pointC.x + b;
I have xy co-ordinate like (200,200). I know the angle calculation from the origin the ball throws. How can I find the initial velocity to reach that particular xy co-ordinate when ball is thrown in 2d Environment?
Iam using
x = v0cosq0t;
y = v0sinq0t - (1/2)gt2.
but time is needed. Without time can I do it? any help please?
I'm assuming that you want the ball to hit that specific point (200,200) at the apex of its path. Well, my physics is a bit rusty, but this is what I've thrown together:
v_y = square_root(2*g*y),
where g is a positive number reflecting the acceleration due to gravity, and y being how high you want to go (200 in this case).
v_x = (x*g) / v_y,
where x is how far in the x direction you want to go (200 in this case), g is as before, and Vy is the answer we got in the previous equation.
These equations remove the need for an angle. However, if you'd rather have the velocity + angle, that's simple:
v0 = square_root(v_x^2 + v_y^2)
and
angle = arctan(v_y / v_x).
Here is the derivation, if you're interested:
(1/2)at^2 + v_yt + 0 = y
(1/2)at^2 + v_yt - y = 0
by quadratic formula,
t = (-v_y +/- square_root(v_y^2 - 2ay)) / a
we also have another equation, because at the apex the vertical velocity is 0:
0 = v_y + at
substitute:
0 = v_y + (-v_y +/- square_root(v_y^2 - 2ay))
0 = square_root(v_y^2 - 2ay)
0 = v_y^2 - 2ay
v_y = square_root(-2ay), or
v_y = square_root(2gy)
For v_x:
v_x*t = x
from before, t = v_y / a, so
v_x = (x*g)/v_y
I hope that made enough sense.
Im sure you can assume the velocity change is instantaneous. Games physics always has some 'dodgy' parts in it because it is too computationally expensive or not important enough to get right down the low granularity information.
You can start the velocity ass instantaneous, and then using a timer class to measure then time between each frame (very rough way of doing it), or you can have a timer class set up in an update loop that will update the physics every x seconds.