Merge pull request #235 from Hemanth21k/q78_edit

Q.78 solution explained with reference

Author: rougier

source/exercises100.ktx

@@ -1090,18 +1090,35 @@ Consider 2 sets of points P0,P1 describing lines (2d) and a point p, how to comp
 No hints provided...
 
 < a78
-def distance(P0, P1, p):
+P0 = np.random.uniform(-10,10,(10,2))
+P1 = np.random.uniform(-10,10,(10,2))
+p  = np.random.uniform(-10,10,( 1,2))
+
+def distance_faster(P0,P1,p):
+    #Author: Hemanth Pasupuleti
+    #Reference: https://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html
+
+    v = P1- P0 
+    v[:,[0,1]] = v[:,[1,0]]
+    v[:,1]*=-1 
+    norm = np.linalg.norm(v,axis=1)   
+    r = P0 - p
+    d = np.abs(np.einsum("ij,ij->i",r,v)) / norm 
+
+    return d
+
+print(distance_faster(P0, P1, p))
+
+##--------------- OR ---------------##
+def distance_slower(P0, P1, p):
     T = P1 - P0
     L = (T**2).sum(axis=1)
     U = -((P0[:,0]-p[...,0])*T[:,0] + (P0[:,1]-p[...,1])*T[:,1]) / L
     U = U.reshape(len(U),1)
     D = P0 + U*T - p
     return np.sqrt((D**2).sum(axis=1))
 
-P0 = np.random.uniform(-10,10,(10,2))
-P1 = np.random.uniform(-10,10,(10,2))
-p  = np.random.uniform(-10,10,( 1,2))
-print(distance(P0, P1, p))
+print(distance_slower(P0, P1, p))
 
 < q79
 Consider 2 sets of points P0,P1 describing lines (2d) and a set of points P, how to compute distance from each point j (P[j]) to each line i (P0[i],P1[i])? (★★★)