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I'm trying to write an algorithm to determine if point is located inside a triangle or on it's edge in 3D coordinate space.
For example, I try to reach such results for different cases
I've figured out how to check if point P inside the triangle, I calculated normal vectors for triangles ABP, BCP, CAP and checked if they are similar.
Can someone explain how to check if a point is on the edge of a triangle (but not outside of a triangle)? You can provide formulas or code as you wish.
Make vectors:
r = p - A (r.x = p.x - A.x, r.y = p.y - A.y, r.z = p.z - A.z)
s = B - A
q = C - A
Calculate normal to ABC plane:
n = s x q (vector product)
Check if p lies in ABC plane using dot product:
dp = n.dot.r
If dp is zero (or has very small value like 1.0e-10 due to the floating point errors, then p is in the plane, and we can continue
Decompose vector p by base vectors s and q. At first check if z-component of normal (n.z) is non-zero. If so, use the next pair of equations (otherwise choose equations for x/z or y/z components):
px = a * sx + b * qx
py = a * sy + b * qy
Solve this system
a = (sy * qx - sx * qy) / (py * qx - px * qy)
b = (px - a * sx) / qx
If resulting coefficients a and b fulfill limits:
a >= 0
b >= 0
a + b <= 1.0
then point p lies in triangle plane inside it.
I have been trying to get the common tangents to two rotated ellipse. I was following the method given by Edward Doolittle in the following thread. The two ellipses are given as the equation given in Wiki.
In the Matrix form the ellipse can be shown as this:
First ellipse is centered at (0,0) rotated by 45 degrees with semi-major and semi-minor axes length as 2,1. Second ellipse is centered at (15,0), rotated by 120 degrees with semi-major and semi-minor axes length as 3,1
Linear combination of the adjoint matrices of the two ellipses are per dual of two ellipse combined
I am getting this value
.
Then I tried to find to find the value of t which will make the conic (above matrix) degenerate.
I found the value of t to be (-0.05,0.29,2.46). However, when I put these values back into the above matrix I am not able to reduce the matrix to two variables form. I am always dealing with 3 variables. For example, if I put t = -0.05 then I get the following:
Can someone please help me with this?
It boils down to finding an algorithm for solving a system of two quadratic equations of two variables, by interpreting it as a projective geometry pencil of conics, then finding the three degenerate conics of the pencil together with a projective transformation that simplifies these three degenerate conics to the point of your being able to read off the solutions very easily in new simplifying coordinate system, and after that transforming them back to the original coordinate system.
I sketched an algorithm in python, I think it seems to work on your example... but I was in a hurry and did not check it properly, so there may be bugs...
import numpy as np
import math
# technical module, functions, details
def homogenize(x):
return np.array([x[0], x[1], 1])
def cos_sin(angle_deg):
return math.cos(angle_deg*math.pi/180), math.sin(angle_deg*math.pi/180)
def rotation(cs_sn):
return np.array([[cs_sn[0], -cs_sn[1]],
[cs_sn[1], cs_sn[0]]])
# defining the isometry (the rotation plus translation) transformation
# between the coordinate system aligned with the conic and a general (world)
# coordinate system
def isom_inverse(angle, translation):
'''
isometry from conic-aligned coordinate system (conic attached)
to global coordinate system (world system)
'''
cos_, sin_ = cos_sin(angle)
return np.array([[cos_, -sin_, translation[0]],
[sin_, cos_, translation[1]],
[ 0, 0, 1]])
def isom(angle, translation):
'''
isometry from global coordinate system (world system)
to conic-aligned coordinate system (conic attached)
'''
cos_, sin_ = cos_sin(-angle)
tr = - rotation((cos_, sin_)).dot(translation)
return np.array([[ cos_, -sin_, tr[0]],
[ sin_, cos_, tr[1]],
[ 0, 0, 1 ]])
# calculating the coinc defined by a pair of axes' lengts,
# axes rotation angle and center of the conic
def Conic(major, minor, angle, center):
D = np.array([[minor**2, 0, 0],
[ 0, major**2, 0],
[ 0, 0, -(minor*major)**2]])
U = isom(angle, center)
return (U.T).dot(D.dot(U))
# calculating the coinc dual to the conic defined by a pair of axes' lengths,
# axes rotation angle and center of the conic
def dual_Conic(major, minor, angle, center):
D_1 = np.array([[major**2, 0, 0],
[ 0, minor**2, 0],
[ 0, 0, -1]])
U_1 = isom_inverse(angle, center)
return (U_1).dot(D_1.dot(U_1.T))
# transforming the matrix of a conic into a vector of six coefficients
# of a quadratic equation with two variables
def conic_to_equation(C):
'''
c[0]*x**2 + c[1]*x*y + c[2]*y**2 + c[3]*x + c[4]*y + c[5] = 0
'''
return np.array([C[0,0], 2*C[0,1], C[1,1], 2*C[0,2], 2*C[1,2], C[2,2]])
# transforming the vector of six coefficients
# of a quadratic equation with two variables into a matrix of
# the corresponding conic
def equation_to_conic(eq):
'''
eq[0]*x**2 + eq[1]*x*y + eq[2]*y**2 + eq[3]*x + eq[4]*y + eq[5] = 0
'''
return np.array([[2*eq[0], eq[1], eq[3]],
[ eq[1], 2*eq[2], eq[4]],
[ eq[3], eq[4], 2*eq[5]]]) / 2
# given a point (x,y) define the vector (x^2, xy, y^2, x, y, 1)
def argument(x):
return np.array([x[0]**2, x[0]*x[1], x[1]**2, x[0], x[1], 1])
# given x = (x[0],x[1]) calculate the value of the quadratic equation with
# six coefficients coeff
def quadratic_equation(x, coeff):
'''
coeff[0]*x**2 + coeff[1]*x*y + coeff[2]*y**2 + coeff[3]*x + coeff[4]*y + coeff[5] = 0
'''
return coeff.dot( argument(x) )
# given a pair of conics, as a pair of symmetric matrices,
# calculate the vector k = (k[0], k[1], k[2]) of values for each of which
# the conic c1 - k[i]*c2 from the pencil of conics c1 - t*c2
# is a degenerate conic (the anti-symmetric product of a pair of linear forms)
# and also find the matrix U
# of the projective transformation that simplifies the geometry of
# the pair of conics, the geometry of the pencil c1 - t*c2 in general,
# as well as the geometry of the three degenerate conics in particular
def transform(c1, c2):
'''
c1 and c2 are 3 by 3 symmetric matrices of the two conics
'''
c21 = np.linalg.inv(c2).dot(c1)
k, U = np.linalg.eig(c21)
return k, U
# the same as before, but for a pair of equations instead of matrices of conics
def eq_transform(eq1, eq2):
'''
eq1 and eq2 = np.array([eq[0], eq[1], eq[2], eq[3], eq[4], eq[5]])
'''
C1 = equation_to_conic(eq1)
C2 = equation_to_conic(eq2)
return transform(C1, C2)
# realizing the matrix U as a projective transformation
def proj(U, x):
if len(x) == 2:
x = homogenize(x)
y = U.dot(x)
y = y / y[2]
return y[0:2]
# find the common points, i.e. points of intersection of a pair of conics
# represented by a pair of symmetric matrices
def find_common_points(c1, c2):
k, U = transform(c1, c2)
L1 = (U.T).dot((c1 - k[0]*c2).dot(U))
L2 = (U.T).dot((c1 - k[1]*c2).dot(U))
sol = np.empty((4,3), dtype=float)
for i in range(2):
for j in range(2):
sol[i+2*j,0:2] = np.array([math.sqrt(abs(L2[2,2] / L2[0,0]))*(-1)**i, math.sqrt(abs(L1[2,2] / L1[1,1]))*(-1)**j])
sol[i+2*j,0:2] = proj(U, sol[i+2*j,0:2])
sol[:,2] = np.ones(4)
return sol
# find the solutions, i.e. the points x=(x[0],x[1]) saisfying the pair
# of quadratic equations
# represented by a pair of vectors eq1 and eq2 of 6 coefficients
def solve_eq(eq1, eq2):
conic1 = equation_to_conic(eq1)
conic2 = equation_to_conic(eq2)
return find_common_points(conic1, conic2)
'''
Esample of finding the common tangents of a pair of conics:
conic 1: major axis = 2, minor axis = 1, angle = 45, center = (0,0)
conic 2: major axis = 3, minor axis = 1, angle = 120, center = (15,0)
'''
a = 2
b = 1
cntr = np.array([0,0])
w = 45
Q1 = Conic(a, b, w, cntr)
dQ1 = dual_Conic(a, b, w, cntr)
a = 3
b = 1
cntr = np.array([15,0])
w = 120
Q2 = Conic(a, b, w, cntr)
dQ2 = dual_Conic(a, b, w, cntr)
R = find_common_points(dQ1, dQ2)
print('')
print(R)
print('')
print('checking that the output forms common tangent lines: ')
print('')
print('conic 1: ')
print(np.diagonal(R.dot(dQ1.dot(R.T))) )
print('')
print('conic 2: ')
print(np.diagonal(R.dot(dQ2.dot(R.T))) )
#conic_to_equation(dQ1)
Some Explanations: Assume you want to find the intersection points of two conics C1 and C2. Let us assume for simplicity that they are real ellipses that intersect in four different points (to avoid complex numbers)
In the case of finding the common tangents to a pair of conics,
simply convert the two conics two corresponding duals and then find the intersection points
of the duals. These intersection points are the equation coefficients of the tangents of the original conics.
There are possibly several different geometric interpretations of this problem, but let us go with the pencil of conics. The two conics C1 and C2 are represented by 3 by 3 symmetric matrices with non-zero determinants, which I have denoted C1 and C2. The linear combination, called pencil of conics generated by C1 and C2, is the t-parametrized family of conics C1 - t*C2 , where t is just a number. What is crucial is that every conic C1 - tC2 passes through the intersection points of C1 and C2 and these are the only four points they all have in common. You can prove this by observing that if x.T * C1 * x = x.T * C1 * x = 0 then
x.T * (C1 - t*C2) * x = x.T * C1 * x - t * x.T * C2 * x = 0 - t*0 = 0. Furthermore, if you take an intersection point of C1 and C1 - t*C2, then C2 = C1 - t*C2 + s*C2 you can apply the same argument when s = t.
In this family of conics, there are three degenerate conics, that are geometrically three pairs of lines. They occur exactly when t is such that det( C1 - t*C2 ) = 0. This is a polynomial equation of degree 3 with respect to t, so there are three, say different solutions k[0], k[1], k[2], due to the niceness of the C1 and C2. Projectively speaking, each degenerate conic C1 - k[j]*C2 is a pair of lines and they have a common intersection point u[:,j] = [ u[0,j] : u[1,j] : u[2,j] ]. Moreover, rank(C1 - k[j]*C2) = 2, so ker(C1 - k[j]*C2) = 1. This point u[:,j] is characterized as a solution to the equation
(C1 - k[j]*C2) * u[:,j] = 0.
Since C2 is invertible (non-degenerate), multiply both sides of the equation by inverse(C2) and obtain the equivalent equation ( (inverse(C2) * C1) - k[j]*Identity ) * u[:,j] = 0 which is an eigenvalue equation, with k[j] as eigenvalue and u[:,j] as eigenvector. The output of the function transform() is the 1 by 3 array k of eigenvalues and the 3 by 3 matrix U = [ u[:,0], u[:,1], u[:,2] ] of eigenvectors.
Conjugating C1 - k[j]*C2 by U, i.e. (U.T)*(C1 - k[j]*C2)*U, is geometrically equivalent to performing a projective transformation that sends u[:,0] and u[:,1] to infinity and u[:,2] to the origin. Thus, U changes the coordinate system from a general one to a special coordinate system, in which two of the degenerate conics are given by pairs of parallel lines and combined they intersect in a rectangle. The third conic is represeneted by the diagnols of the rectangle.
In this new picture, the intersection problem can be easily solved, just by reading off one solution from the entries of the matrices (U.T)*(C1 - k[j]*C2)*U (the intersection points are the vertices of the rectangle, so if you find one of them, the others are simply mirror symmetric of each other).
This question already has answers here:
Finding whether a point lies inside a rectangle or not
(10 answers)
Closed 2 years ago.
I have a rotated rectangle with these coordinates as vertices:
1 670273 4879507
2 677241 4859302
3 670388 4856938
4 663420 4877144
And I have points with these coordinates:
670831 4867989
675097 4869543
Using only the Python 2.7 standard library, I want to determine if the points fall within the rotated rectangle.
I am not able to add additional Python libraries to my Jython implementation
What would it take to do this?
A line equation of the form ax+by+c==0 can be constructed from 2 points. For a given point to be inside a convex shape, we need testing whether it lies on the same side of every line defined by the shape's edges.
In pure Python code, taking care of writing the equations avoiding divisions, this could be as follows:
def is_on_right_side(x, y, xy0, xy1):
x0, y0 = xy0
x1, y1 = xy1
a = float(y1 - y0)
b = float(x0 - x1)
c = - a*x0 - b*y0
return a*x + b*y + c >= 0
def test_point(x, y, vertices):
num_vert = len(vertices)
is_right = [is_on_right_side(x, y, vertices[i], vertices[(i + 1) % num_vert]) for i in range(num_vert)]
all_left = not any(is_right)
all_right = all(is_right)
return all_left or all_right
vertices = [(670273, 4879507), (677241, 4859302), (670388, 4856938), (663420, 4877144)]
The following plot tests the code visually for several shapes. Note that for shapes with horizontal and vertical lines usual line equations could provoke division by zero.
import matplotlib.pyplot as plt
import numpy as np
vertices1 = [(670273, 4879507), (677241, 4859302), (670388, 4856938), (663420, 4877144)]
vertices2 = [(680000, 4872000), (680000, 4879000), (690000, 4879000), (690000, 4872000)]
vertices3 = [(655000, 4857000), (655000, 4875000), (665000, 4857000)]
k = np.arange(6)
r = 8000
vertices4 = np.vstack([690000 + r * np.cos(k * 2 * np.pi / 6), 4863000 + r * np.sin(k * 2 * np.pi / 6)]).T
all_shapes = [vertices1, vertices2, vertices3, vertices4]
for vertices in all_shapes:
plt.plot([x for x, y in vertices] + [vertices[0][0]], [y for x, y in vertices] + [vertices[0][1]], 'g-', lw=3)
for x, y in zip(np.random.randint(650000, 700000, 1000), np.random.randint(4855000, 4880000, 1000)):
color = 'turquoise'
for vertices in all_shapes:
if test_point(x, y, vertices):
color = 'tomato'
plt.plot(x, y, '.', color=color)
plt.gca().set_aspect('equal')
plt.show()
PS: In case you are running a 32-bit version of numpy, with this size of integers it might be necessary to convert the values to float to avoid overflow.
If this calculation needs to happen very often, the a,b,c values can be precalculated and stored. If the direction of the edges is known, only one of all_left or all_right is needed.
When the shape is fixed, a text version of the function can be generated:
def generate_test_function(vertices, is_clockwise=True, function_name='test_function'):
ext_vert = list(vertices) + [vertices[0]]
unequality_sign = '>=' if is_clockwise else '<='
print(f'def {function_name}(x, y):')
parts = []
for (x0, y0), (x1, y1) in zip(ext_vert[:-1], ext_vert[1:]):
a = float(y1 - y0)
b = float(x0 - x1)
c = a * x0 + b * y0
parts.append(f'({a}*x + {b}*y {unequality_sign} {c})')
print(' return', ' and '.join(parts))
vertices = [(670273, 4879507), (677241, 4859302), (670388, 4856938), (663420, 4877144)]
generate_test_function(vertices)
This would generate a function as:
def test_function(x, y):
return (-20205.0*x + -6968.0*y >= -47543270741.0) and (-2364.0*x + 6853.0*y >= 31699798882.0) and (20206.0*x + 6968.0*y >= 47389003912.0) and (2363.0*x + -6853.0*y >= -31855406372.0)
This function then can be copy-pasted and optimized by the Jython compiler. Note that the shape doesn't need to be rectangular. Any convex shape will do, allowing to use a tighter box.
Take three consequent vertices A, B, C (your 1,2,3)
Find lengths of sides AB and BC
lAB = sqrt((B.x - A.x)^2+(B.y - A.y)^2)
Get unit (normalized) direction vectors
uAB = ((B.x - A.x) / lAB, (B.y - A.y) / lAB)
For tested point P get vector BP
BP = ((P.x - B.x), (P.y - B.y))
And calculate signed distances from sides to point using cross product
SignedDistABP = Cross(BP, uAB) = BP.x * uAB.y - BP.y * uAB.x
SignedDistBCP = - Cross(BP, uBC) = - BP.x * uBC.y + BP.y * uBC.x
For points inside rectangle both distances should have the same sign - either negative or positive depending on vertices order (CW or CCW), and their absolute values should not be larger than lBC and lAB correspondingly
Abs(SignedDistABP) <= lBC
Abs(SignedDistBCP) <= lAB
As the shape is an exact rectangle, the easiest is to rotate all points by the angle
-arctan((4859302-4856938)/(677241-670388))
Doing so, the rectangle becomes axis-aligned and you just have to perform four coordinate comparisons. Rotations are easy to compute with complex numbers.
In fact you can simply represent all points as complex numbers, compute the vector defined by some side, and multiply everything by the conjugate.
A slightly different approach is to consider the change of coordinate frame that brings some corner to the origin and two incident sides to (1,0) and (0,1). This is an affine transformation. Then your test boils down to checking insideness to the unit square.
I would like to fit a 2D array by an elliptic function: (x / a)² + (y / b)² = 1 ----> (and so get the a and b)
And then, be able to replot it on my graph.
I found many examples on internet, but no one with this simple Cartesian equation. I probably have searched badly ! I think a basic solution for this problem could help many people.
Here is an example of the data:
Sadly, I can not put the values... So let's assume that I have an X,Y arrays defining the coordinates of each of those points.
This can be solved directly using least squares. You can frame this as minimizing the sum of squares of quantity (alpha * x_i^2 + beta * y_i^2 - 1) where alpha is 1/a^2 and beta is 1/b^2. You have all the x_i's in X and the y_i's in Y so you can find the minimizer of ||Ax - b||^2 where A is an Nx2 matrix (i.e. [X^2, Y^2]), x is the column vector [alpha; beta] and b is column vector of all ones.
The following code solves the more general problem for an ellipse of the form Ax^2 + Bxy + Cy^2 + Dx +Ey = 1 though the idea is exactly the same. The print statement gives 0.0776x^2 + 0.0315xy+0.125y^2+0.00457x+0.00314y = 1 and the image of the ellipse generated is also below
import numpy as np
import matplotlib.pyplot as plt
alpha = 5
beta = 3
N = 500
DIM = 2
np.random.seed(2)
# Generate random points on the unit circle by sampling uniform angles
theta = np.random.uniform(0, 2*np.pi, (N,1))
eps_noise = 0.2 * np.random.normal(size=[N,1])
circle = np.hstack([np.cos(theta), np.sin(theta)])
# Stretch and rotate circle to an ellipse with random linear tranformation
B = np.random.randint(-3, 3, (DIM, DIM))
noisy_ellipse = circle.dot(B) + eps_noise
# Extract x coords and y coords of the ellipse as column vectors
X = noisy_ellipse[:,0:1]
Y = noisy_ellipse[:,1:]
# Formulate and solve the least squares problem ||Ax - b ||^2
A = np.hstack([X**2, X * Y, Y**2, X, Y])
b = np.ones_like(X)
x = np.linalg.lstsq(A, b)[0].squeeze()
# Print the equation of the ellipse in standard form
print('The ellipse is given by {0:.3}x^2 + {1:.3}xy+{2:.3}y^2+{3:.3}x+{4:.3}y = 1'.format(x[0], x[1],x[2],x[3],x[4]))
# Plot the noisy data
plt.scatter(X, Y, label='Data Points')
# Plot the original ellipse from which the data was generated
phi = np.linspace(0, 2*np.pi, 1000).reshape((1000,1))
c = np.hstack([np.cos(phi), np.sin(phi)])
ground_truth_ellipse = c.dot(B)
plt.plot(ground_truth_ellipse[:,0], ground_truth_ellipse[:,1], 'k--', label='Generating Ellipse')
# Plot the least squares ellipse
x_coord = np.linspace(-5,5,300)
y_coord = np.linspace(-5,5,300)
X_coord, Y_coord = np.meshgrid(x_coord, y_coord)
Z_coord = x[0] * X_coord ** 2 + x[1] * X_coord * Y_coord + x[2] * Y_coord**2 + x[3] * X_coord + x[4] * Y_coord
plt.contour(X_coord, Y_coord, Z_coord, levels=[1], colors=('r'), linewidths=2)
plt.legend()
plt.xlabel('X')
plt.ylabel('Y')
plt.show()
Following the suggestion by ErroriSalvo, here is the complete process of fitting an ellipse using the SVD. The arrays x, y are coordinates of the given points, let's say there are N points. Then U, S, V are obtained from the SVD of the centered coordinate array of shape (2, N). So, U is a 2 by 2 orthogonal matrix (rotation), S is a vector of length 2 (singular values), and V, which we do not need, is an N by N orthogonal matrix.
The linear map transforming the unit circle to the ellipse of best fit is
sqrt(2/N) * U * diag(S)
where diag(S) is the diagonal matrix with singular values on the diagonal. To see why the factor of sqrt(2/N) is needed, imagine that the points x, y are taken uniformly from the unit circle. Then sum(x**2) + sum(y**2) is N, and so the coordinate matrix consists of two orthogonal rows of length sqrt(N/2), hence its norm (the largest singular value) is sqrt(N/2). We need to bring this down to 1 to have the unit circle.
N = 300
t = np.linspace(0, 2*np.pi, N)
x = 5*np.cos(t) + 0.2*np.random.normal(size=N) + 1
y = 4*np.sin(t+0.5) + 0.2*np.random.normal(size=N)
plt.plot(x, y, '.') # given points
xmean, ymean = x.mean(), y.mean()
x -= xmean
y -= ymean
U, S, V = np.linalg.svd(np.stack((x, y)))
tt = np.linspace(0, 2*np.pi, 1000)
circle = np.stack((np.cos(tt), np.sin(tt))) # unit circle
transform = np.sqrt(2/N) * U.dot(np.diag(S)) # transformation matrix
fit = transform.dot(circle) + np.array([[xmean], [ymean]])
plt.plot(fit[0, :], fit[1, :], 'r')
plt.show()
But if you assume that there is no rotation, then np.sqrt(2/N) * S is all you need; these are a and b in the equation of the ellipse.
You could try a Singular Value Decomposition of the data matrix.
https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.linalg.svd.html
First center the data by subtracting mean values of X,Y from each column respectively.
X=X-np.mean(X)
Y=Y-np.mean(Y)
D=np.vstack(X,Y)
Then, apply SVD and extract
-eigenvalues (members of s) -> axis length
-eigenvectors(U) -> axis orientation
U, s, V = np.linalg.svd(D, full_matrices=True)
This should be a least-squares fit.
Of course, things can get more complicated than this, please see
https://www.emis.de/journals/BBMS/Bulletin/sup962/gander.pdf
I have this algorithm here:
pc = # the point you are coloring now
p0 = # start point
p1 = # end point
v = p1 - p0
d = Length(v)
v = Normalize(v) # or Scale(v, 1/d)
v0 = pc - p0
t = Dot(v0, v)
t = Clamp(t/d, 0, 1)
color = (start_color * t) + (end_color * (1 - t))
to generate point to point linear gradients. It works very well for me. I was wondering if there was a similar algorithm to generate radial gradients. By similar, I mean one that solves for color at point P rather than solve for P at a certain color (where P is the coordinate you are painting).
Thanks
//loop through vector
//x and y px position
int x = i%w;
int y = i/w;
float d = distance(center,int2(x,y));
//if within the grad circle
if(d < radius)
{
//somehow set v[i] alpha to this:
float a = d/r;
}
Linerise over atan2(dy,dx) where dx is x-center, and dy is y-center.
cx # center x
cy # center y
r1 # ring is defined by two radius
r2 # r1 < r2
c1 # start color
c2 # stop color
ang # start angle
px # currect point x,y
py
if( px^2 + py^2 <= r2^2 AND px^2 + py^2 >= r1^2 ) # lies in ring?
t= atan2(py-cy,px-cx)+ang
t= t+ pi # atan2 is from -pi to pi
if (t > 2* pi) # it might over 2pi becuse of +ang
t=t-2*pi
t=t/(2*pi) # normalise t from 0 to 1
color = (c1 * t) + (c2 * (1 - t))
Problem whit this algorhitm is that ang is actualy wrong and should be rotated by pi and normalized between 0 and 2pi.
Based on the comment, what you want can still be viewed as a linear gradient -- i.e. you have a line from the center to the outside of the circle, and you have a linear gradient along that line. As such, the calculation is virtually identical to what you already had.
Edit: Okay, apparently I misunderstood what you want. To figure a gradient running around a radius, you still basically linearize it -- figure out the circumference at that radius (2*Pi*R), and then do a linear interpolation along a line of that length.