Given a line passing through two points A and B and an arbitrary point C in a 3-D plane, the task is to find the shortest distance between the point C and the line passing through the points A and B.
Input: A = (5, 2, 1), B = (3, 1, -1), C = (0, 2, 3) Output: Shortest Distance is 5 Input: A = (4, 2, 1), B = (3, 2, 1), C = (0, 2, 0) Output: Shortest Distance is 1
Consider a point C and a line that passes through A and B as shown in the below figure.
Now Consider the vectors, AB and AC and the shortest distance as CD. The Shortest Distance is always the perpendicular distance. The point D is taken on AB such that CD is perpendicular to AB.
Construct BP and CP as shown in the figure to form a Parallelogram. Now C is a vertex of parallelogram ABPC and CD is perpendicular to Side AB. Hence CD is the height of the parallelogram.
Note: In the case when D does not fall on line segment AB there will be a point D’ such that PD’ is perpendicular to AB and D’ lies on line segment AB with CD = PD’.
The magnitude of cross product AB and AC gives the Area of the parallelogram. Also, the area of a parallelogram is Base * Height = AB * CD. So,
CD = |ABxAC| / |AB|
Below is the CPP program to find the shortest distance:
Shortest Distance is : 1.63299
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Distance between a point and a Plane in 3 D
- Find foot of perpendicular from a point in 2 D plane to a Line
- Shortest distance between a point and a circle
- Perpendicular distance between a point and a Line in 2 D
- Maximum distance between two points in coordinate plane using Rotating Caliper's Method
- Hammered distance between N points in a 2-D plane
- Equation of straight line passing through a given point which bisects it into two equal line segments
- Minimum distance from a point to the line segment using Vectors
- Find mirror image of a point in 2-D plane
- Number of jump required of given length to reach a point of form (d, 0) from origin in 2D plane
- Mirror of a point through a 3 D plane
- Find the foot of perpendicular of a point in a 3 D plane
- Check if a line at 45 degree can divide the plane into two equal weight parts
- Count of intersections of M line segments with N vertical lines in XY plane
- Find the shortest distance between any pair of two different good nodes
- Ratio of the distance between the centers of the circles and the point of intersection of two direct common tangents to the circles
- Ratio of the distance between the centers of the circles and the point of intersection of two transverse common tangents to the circles
- Distance of chord from center when distance between center and another equal length chord is given
- Shortest distance to every other character from given character
- Shortest distance from the centre of a circle to a chord
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.