Shortest Path Calculator for Physics

Calculator Inputs

Calculation Mode

Choose planar motion or full spatial motion.

Coordinate Unit

All coordinates use this unit.

Speed Unit

Used only for travel time.

Start X

Start Y

Start Z

End X

End Y

End Z

Average Speed

Enter zero to skip time calculations.

Compare with waypoint path

Waypoint X

Waypoint Y

Waypoint Z

Example Data Table

This example compares a direct route with an indirect waypoint route using 2D coordinates and constant speed.

Mode	Start Point	End Point	Waypoint	Coordinate Unit	Average Speed
2D	(0, 0)	(10, 8)	(4, 3)	m	5 m/s
3D	(0, 0, 0)	(7, 9, 4)	(3, 4, 2)	m	6 m/s

Formula Used

2D shortest path:
d = √[(x₂ − x₁)² + (y₂ − y₁)²]

3D shortest path:
d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]

Waypoint route:
d_waypoint = d(start, waypoint) + d(waypoint, end)

Travel time:
t = distance ÷ speed

2D heading angle:
θ = atan2(Δy, Δx)

3D elevation angle:
φ = atan2(Δz, √(Δx² + Δy²))

In physics, the direct straight-line displacement between two points is the shortest possible route in uniform, obstacle-free space.

How to Use This Calculator

Select 2D or 3D mode.
Choose the coordinate unit for all entered values.
Enter the start and end coordinates.
Optionally enable waypoint comparison to test an indirect route.
Enter average speed if you want travel time.
Press Calculate Shortest Path to show the result above the form.
Review the graph, comparison table, and export the results as CSV or PDF.

FAQs

1. What does this shortest path calculator measure?

It measures the straight-line displacement between two points in 2D or 3D space. It can also compare that shortest route with an optional waypoint path and estimate travel time from average speed.

2. Why is the direct path considered shortest?

In Euclidean space, the straight segment connecting two points has the minimum distance. Any detour, bend, or waypoint adds length unless it lies exactly on the same direct line.

3. What is the difference between path length and displacement?

Displacement is the direct vector from start to end. Path length is the actual route traveled. If motion includes detours, the path length becomes greater than the displacement magnitude.

4. When should I use 3D mode?

Use 3D mode when height or depth matters, such as particle motion, drone trajectories, lab positioning, or spatial geometry problems. Use 2D mode for flat plane motion.

5. What does the heading angle mean?

The heading angle is the direction of the displacement vector measured from the positive x-axis in the x-y plane. It helps describe orientation, not just distance.

6. How is travel time calculated?

Travel time is found by dividing distance by speed after unit conversion. The calculator converts your chosen speed into meters per second internally for consistent physics calculations.

7. What are direction cosines?

Direction cosines are the normalized x, y, and z components of the displacement vector. They show how strongly the motion points along each axis in 3D space.

8. Can this calculator handle obstacles or curved paths?

This version focuses on straight-line geometry and a single optional waypoint comparison. It does not solve obstacle-avoidance networks, refraction paths, or full optimization over curved surfaces.