# edge_edge_distance

## `edge_edge_distance(P1, Q1, P2, Q2)`

Computes the distance between two edges (segments) in 3D space.

Parameters:

Name Type Description Default
`P1` `(3,) numpy array`

start point of first edge

required
`Q1` `(3,) numpy array`

end point of first edge

required
`P2` `(3,) numpy array`

start point of second edge

required
`Q2` `(3,) numpy array`

end point of second edge

required

Returns:

Name Type Description
`d` `float`

The distance between the two edges.

`R1` `(3,) numpy array`

The closest point on the first edge to the second edge.

`R2` `(3,) numpy array`

The closest point on the second edge to the first edge.

#### Notes

This function is based on the algorithm from the "Proximity Query Pack" by Eric Larsen and Stefan Gottschalk

Examples:

``````import numpy as np
from gpytoolbox import edge_edge_distance
P1 = np.array([0.0,0.0,0.0])
P2 = np.array([1.0,0.0,0.0])
Q1 = np.array([0.0,1.0,0.0])
Q2 = np.array([1.0,1.0,0.0])
dist,R1,R2 = gpytoolbox.edge_edge_distance(P1,Q1,P2,Q2)
``````
Source code in `src/gpytoolbox/edge_edge_distance.py`
 ``` 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197``` ``````def edge_edge_distance(P1,Q1,P2,Q2): """ Computes the distance between two edges (segments) in 3D space. Parameters ---------- P1 : (3,) numpy array start point of first edge Q1 : (3,) numpy array end point of first edge P2 : (3,) numpy array start point of second edge Q2 : (3,) numpy array end point of second edge Returns ------- d : float The distance between the two edges. R1 : (3,) numpy array The closest point on the first edge to the second edge. R2 : (3,) numpy array The closest point on the second edge to the first edge. Notes ----- This function is based on the algorithm from the "Proximity Query Pack" by Eric Larsen and Stefan Gottschalk Examples -------- ```python import numpy as np from gpytoolbox import edge_edge_distance P1 = np.array([0.0,0.0,0.0]) P2 = np.array([1.0,0.0,0.0]) Q1 = np.array([0.0,1.0,0.0]) Q2 = np.array([1.0,1.0,0.0]) dist,R1,R2 = gpytoolbox.edge_edge_distance(P1,Q1,P2,Q2) ``` """ P = P1 Q = P2 A = Q1 - P1 B = Q2 - P2 # VmV(T,Q,P); # P -> P1 # Q -> P2 T = Q - P # A_dot_A = VdotV(A,A); # B_dot_B = VdotV(B,B); # A_dot_B = VdotV(A,B); # A_dot_T = VdotV(A,T); # B_dot_T = VdotV(B,T); A_dot_A = np.dot(A,A) B_dot_B = np.dot(B,B) A_dot_B = np.dot(A,B) A_dot_T = np.dot(A,T) B_dot_T = np.dot(B,T) # // t parameterizes ray P,A # // u parameterizes ray Q,B # PQP_REAL t,u; # // compute t for the closest point on ray P,A to # // ray Q,B # PQP_REAL denom = A_dot_A*B_dot_B - A_dot_B*A_dot_B; # t = (A_dot_T*B_dot_B - B_dot_T*A_dot_B) / denom; # // clamp result so t is on the segment P,A # if ((t < 0) || isnan(t)) t = 0; else if (t > 1) t = 1; # // find u for point on ray Q,B closest to point at t # u = (t*A_dot_B - B_dot_T) / B_dot_B; denom = A_dot_A*B_dot_B - A_dot_B*A_dot_B if denom == 0: t = 0 else: t = (A_dot_T*B_dot_B - B_dot_T*A_dot_B) / denom # print("t: ",t) if((t < 0) or np.isnan(t)): t = 0 elif(t > 1): t = 1 # print("t: ",t) if B_dot_B == 0: u = 0 else: u = (t*A_dot_B - B_dot_T) / B_dot_B # print("u: ",u) # if ((u <= 0) || isnan(u)) { # VcV(Y, Q); # t = A_dot_T / A_dot_A; # if ((t <= 0) || isnan(t)) { # VcV(X, P); # VmV(VEC, Q, P); # } # else if (t >= 1) { # VpV(X, P, A); # VmV(VEC, Q, X); # } # else { # VpVxS(X, P, A, t); # VcrossV(TMP, T, A); # VcrossV(VEC, A, TMP); # } # } if ((u<=0) or np.isnan(u)): Y = Q.copy() t = A_dot_T / A_dot_A if ((t<=0) or np.isnan(t)): X = P.copy() elif (t>=1): X = P + A else: X = P + t*A # else if (u >= 1) { # VpV(Y, Q, B); # t = (A_dot_B + A_dot_T) / A_dot_A; # if ((t <= 0) || isnan(t)) { # VcV(X, P); # VmV(VEC, Y, P); # } # else if (t >= 1) { # VpV(X, P, A); # VmV(VEC, Y, X); # } # else { # VpVxS(X, P, A, t); # VmV(T, Y, P); # VcrossV(TMP, T, A); # VcrossV(VEC, A, TMP); # } # } elif (u>=1): Y = Q + B t = (A_dot_T + A_dot_B) / A_dot_A if ((t<=0) or np.isnan(t)): X = P.copy() elif (t>=1): X = P + A else: X = P + t*A # else { # VpVxS(Y, Q, B, u); # if ((t <= 0) || isnan(t)) { # VcV(X, P); # VcrossV(TMP, T, B); # VcrossV(VEC, B, TMP); # } # else if (t >= 1) { # VpV(X, P, A); # VmV(T, Q, X); # VcrossV(TMP, T, B); # VcrossV(VEC, B, TMP); # } # else { # VpVxS(X, P, A, t); # VcrossV(VEC, A, B); # if (VdotV(VEC, T) < 0) { # VxS(VEC, VEC, -1); # } # } # } else: Y = Q + u*B if ((t<=0) or np.isnan(t)): X = P.copy() elif (t>=1): # print("here") X = P + A else: X = P + t*A R1 = X R2 = Y dist = np.linalg.norm(R1-R2) return dist, R1, R2 ``````