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    <p>Thanks to @ubutbu for pointing out the solution for me, figured I would post a working implementation in case someone googles upon this question with a need in the future:</p> <pre><code>def findParallelT(P, V): """finds the t value along a quadratic Bezier such that its tangent (1st derivative) is parallel with the direction vector V. P : a 3x2 matrix containing the control points of B(t). V : a pair of values representing the direction of interest (magnitude is ignored). returns 0.0 &lt;= t &lt;= 1.0 or None Note the result may be in the same direction or flipped 180 degrees from V""" #refer to answer given by 'unutbu' on the stackoverflow question: #http://stackoverflow.com/questions/20825173/how-to-find-a-point-if-any-on-quadratic-bezier-with-a-given-tangent-direction #for explanation. # also the 'Rearrange It' app at http://www.wolframalpha.com/widgets/view.jsp?id=4be4308d0f9d17d1da68eea39de9b2ce was invaluable. assert len(P)==3 and len(P[0])==2 and len(P[1])==2 and len(P[2])==2 assert len(V)==2 P1=P[0] P2=P[1] P3=P[2] # B(t) = P1 * (1-t)**2 + 2*P2*(1-t)*t + P3*t**2 # B'(t) = 2 * (1-t) * (P2 - P1) + 2*t*(P3-P2) # B'(t) = (a*t + b, c*t + d), V = (e, f) a = -2 * (P2[0] - P1[0]) + 2 * (P3[0]-P2[0]) b = 2 * (P2[0] - P1[0]) c = -2 * (P2[1] - P1[1]) + 2 * (P3[1]-P2[1]) d = 2 * (P2[1] - P1[1]) e = V[0] f = V[1] # -(a*t+b)*f + (c*t+d)*e = 0 for parallel... t = (d*e - b*f) / (a*f - c*e) num = float(d * e - b * f) den = float(a * f - c * e) if den == 0: return None else: t = num / den return t if 0.0 &lt;= t &lt;= 1.0 else None if __name__ == "__main__": assert findParallelT([[0.0,0.0], [0.0,25.0], [25.0,25.0]], [25.0,25.0]) == 0.5 assert findParallelT([[0.0,0.0], [0.0,25.0], [25.0,25.0]], [25.0,-25.0]) == None assert findParallelT([[200.0,200.0], [10.0, 20.0], [300.0, 0.0]], [10.0,0.0]) == None assert findParallelT([[407.5, 376.5],[321.0,463.0],[321.0,586.0]], [-246.0,0.0] ) == None assert findParallelT([[617.0, 882.0],[740.0, 882.0], [826.5, 795.5]], [-245.0,0.0]) == 0.0 </code></pre>
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    1. POHow to find a point (if any) on quadratic Bezier with a given tangent direction?
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