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Calcimator

Motion Path Calculator

Calculate arc length along motion paths between two points.

About this calculator

This calculator estimates the arc length of a curved motion path between two points on screen, using a parabolic approximation rather than a true Bezier or spline curve. You give it a Start X/Y and End X/Y coordinate pair plus an Arc Height — how far the path bows above the straight line connecting the two points — and it returns the resulting path length in pixels, the straight-line distance for comparison, a rough per-frame speed at 24fps, and the departure angle from the start point. The parabolic model treats the arc as two straight legs meeting at the peak, so it works well for the gentle bows typical of UI motion (toasts sliding in, icons popping along a curved trajectory) but will under- or over-state length on paths with sharp direction changes, S-curves, or multiple control points, none of which this simple two-point model can represent.

Raising Arc Height always lengthens the path — the two legs of the approximated triangle both grow — and it never shortens it, whatever the endpoints are. Speed (px/frame @24fps) divides that arc length by a fixed 24 frames, which assumes exactly one second of travel; if your animation timeline uses a different duration or a non-linear easing curve, scale or reinterpret this figure rather than reading it as your actual keyframe spacing. The direction angle read from Start Y and End Y is a straight bearing between the endpoints, not the tangent direction at any point along the curved arc itself.

Inputs

px
px
px
px
px

Results

Arc Length

721.11 px

Straight Distance600 px
Speed (px/frame @24fps)30.05
Direction Angle0°
How to Use This Calculator
  1. Enter Start X, Start Y, and End X.
  2. Set End Y and Arc Height.
  3. Review the Arc Length (px) result.
  4. Use Straight Distance (px) and Speed (px/frame @24fps) to inform your decision.

How the result changes with End X

End XArc Length
350471.7 px
525583.63 px
1,0501,030.78 px
1,7501,697.79 px

What each input means

Start X
X coordinate of the start position.
Start Y
Y coordinate of the start position.
End X
X coordinate of the end position.
End Y
Y coordinate of the end position.
Arc Height
Height of the arc above the straight-line path (0 for straight).

How this is calculated

Worked example, using the default values

  1. Identify Input Parameters
    5 parameters
    Start X = 100, Start Y = 500, End X = 700, End Y = 500, Arc Height = 200 = 5 input(s) provided
  2. Calculate Arc Length
    721.11 = 721.11
  3. Calculate Straight Distance
    Straight Distance
    600 = 600
  4. Calculate Speed
    Speed
    30.05 = 30.05

Engine last updated . Checked against 2 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.

Frequently Asked Questions

Does increasing Arc Height always make the path longer?

Yes. Across the calculator's full 0-5000px Arc Height range, arc length only ever increases or stays flat as you raise the height — it never comes back down. The parabolic approximation treats the arc as two straight legs from each endpoint up to the peak, and both legs necessarily lengthen as the peak moves farther from the straight line, whatever your Start and End coordinates are set to.

Why does the Direction Angle change when I only move End Y?

Direction Angle is the straight bearing from the start point to the end point, measured with atan2, so it always responds to End Y — but which way it moves depends on whether End X sits to the right or left of Start X. When End X is greater than or equal to Start X, raising End Y turns the bearing upward and the angle rises. When End X is less than Start X (a right-to-left path, just as legitimate an input as left-to-right), the relationship mirrors and raising End Y instead turns the angle downward. It is not derived from the curved arc itself, only the two endpoints.

What does the Speed (px/frame @24fps) figure actually assume?

It divides the computed Arc Length by a fixed 24 frames, which is only correct if your motion is meant to complete in exactly one second at 24 frames per second. A different clip duration, a different frame rate, or any eased (non-linear) motion curve will make the real per-frame travel distance vary rather than sit at this one constant figure, so treat it as a rough reference speed rather than a per-frame keyframe value.

Can this calculator represent a true Bezier or spline motion path?

No — it only models a single symmetric arc bowed above the straight line between two endpoints, approximated as two straight legs meeting at the peak. A real Bezier curve with independent control-point handles, an S-curve, or a multi-point spline has a shape this two-point parabolic model cannot capture, so treat the Arc Length here as an estimate for a simple bow, not an exact match for handle-based easing tools.

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