![]() ![]() The curvature of a straight line is zero. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. Smaller circles bend more sharply, and hence have higher curvature. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.įor curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. ![]() In mathematics, curvature is any of several strongly related concepts in geometry. Plt.title('curvature = '.A migrating wild-type Dictyostelium discoideum cell whose boundary is colored by curvature. Y_fit = comp_curv.yc + comp_curv.r*np.sin(theta_fit) X_fit = comp_curv.xc + comp_curv.r*np.cos(theta_fit) Theta_fit = np.linspace(-np.pi, np.pi, 180) The axis corresponding to derivatives must be coherent with the col_deriv option of leastsq"""ĭf_dc = df_dc - df_dc.mean(axis=1)Ĭenter_estimate = np.r_Ĭenter = optimize.leastsq(self.f, center_estimate, Dfun=self.df, col_deriv=True) """ calculate the algebraic distance between the data points and the mean circle centered at c=(xc, yc) """ """ calculate the distance of each 2D points from the center (xc, yc) """ Self.yc = 0 # Y-coordinate of circle center Self.xc = 0 # X-coordinate of circle center Here's the code: from scipy import optimize īased on above's link, I wrote a small python script that shows how you can compute the curvature using a linear least squares fit. However, it is not trivial to find the best circle fit for your data. Still, one can use the same approach: fit a circle through the datapoints and compute the reciprocal of the radius. If more points are available, it would be beneficial to use them. If den = 0.0: # Very unlikely, but just to be sureĪdditional note: this might not be optimal as only three points are used to compute the curvature. K = np.sqrt((a+(b+c))*(c-(a-b))*(c+(a-b))*(a+(b-c))) / 4 # Heron's formula for triangle's surfaceĭen = a*b*c # Denumerator make sure there is no division by zero. ![]() # Compute inverse radius of circle using surface of triangle (for which Heron's formula is used) # points_utm is a 3-by-2 array, containing the easting and northing coordinates of 3 pointsĪ = np.hypot(points_utm - points_utm, points_utm - points_utm)ī = np.hypot(points_utm - points_utm, points_utm - points_utm)Ĭ = np.hypot(points_utm - points_utm, points_utm - points_utm) I used Heron's formula for computing the area of the triangle $k$. I happened to code this in the past in python. Obviously, the curvature is the reciprocal of this, thus Where $a$, $b$ and $c$ denote the distances between the three points and $k$ denotes the area of the triangle formed by the three points. The curvature is the inverse of the radius. As dbx mentioned in the comments, you can use the radius of a circle passing through three points. ![]()
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