Find the distance d(P, l) of a Point P(x, y) from a line l in the 2-dimensional space ℝ2. Line l can be specified in one of the following forms:
Standard ax + by = c⇒ d = | ax + by - c |
√
a2 + b2
Slope-intercep y = mx + b⇒ d = | mx - y + b |
√
m2 + 1
Parametric x = x0 + at, y = y0 + bt⇒ d = | bx - ay - bx0 + ay0 |
√
a2 + b2
Point P:
( , )
Line l
x + y =
y = x +
x = + t y = + t
Distanced(P, l) = 0
Find the distance d(P, l) of a Point P(x, y, z) from a line l in the 3-dimensional space ℝ3. Line l is specified in the parametric form as x = x0 + at, y = y0 + bt, z = z0 + ct Distance is calculate as d(P, l) = ||
OP
→
⨯
v
→
||||
v
→
|| , where O is some point on the line l and
v
→
is directional vector of l (lies on l ) .
Point P:
( , , )
Line l :
x = + t
y = + t
z = + t
Distanced(P, l) = 0
Find the distance d(P, 𝛼) of a Point P(x, y, z) to the plane 𝛼 in the 3-dimensional space ℝ3. Plane 𝛼 is specified by the equation ax + by + cz + d = 0 , where n = (a, b, c) is its normal vector and (x, y, z) ∈ ℝ3
Point P:
( , , )
Plane 𝛼 :
x + y + z + = 0
Distanced(P, 𝛼) = 0
Find the distance d(P, △ABC) of a Point P(x, y, z) to a triangele △ABC formed by the 3 points A, B and C in the 3-dimensional space ℝ3. Distance is defined as the shortest of either the distances of the point P to the triangle's vertices A, B and C, or the distances to the edges AB, BC and CA, if the projection of P on the line containing the edge lies within the edge, or the distance to the plane containing the triangle △ABC, if the projection of P on the plane lies within the triangle.
Point P:
( , , )
△ABC
A :
( , , )
B :
( , , )
C :
( , , )
Distanced(P, △ABC) = 0
Distance is calculated based on point P being closest to A
Find the distance d(P, h) of a Point P(x1, x2, ... , xn) to the hyperplane h in the n-dimensional space ℝn. Hyperplane h is specified by the equation a1x1 + a2x2 + ... + anxn + b = 0 , where (a1, a2, ... , an) is orthogonal vector to the hyperplane, b is some constant and (x1, x2, ... , xn) is an arbitrary point from ℝn Note: First, set up the dimension n. Due to the limited space on the screen, maximum dimension is 30.
