Racing Line Lab

Racing Line Geometry

Shape the path through a 90° corner and measure what the geometry allows.

Track editor · drag T turn-in, A apex, X track-out slow fast
STARTFINISHcorner entrycorner exitR min 44.2 mTRacing line — Turn-in pointARacing line — Apex control pointXRacing line — Track-out point

Line colour shows the curvature-limited speedCurvature-limited speed v_lim = √(μg/|κ|): the fastest steady speed the path’s local bend allows with pure cornering. (49 → 108+ mph). The dashed circle is the osculating circle — its radius is the local radius of curvatureRadius R of the circle that best matches the path at a point (R = 1/κ). Large R means a gentle curve; straights have R → ∞.. Hover the charts to move it. Focused control points also move with the arrow keys.

Racing line preset
1.10
Path measurements
Line length
192.32m
±0.00 m vs preset
Min radiusRadius R of the circle that best matches the path at a point (R = 1/κ). Large R means a gentle curve; straights have R → ∞.
44.2m
Curvature-limited min speed
48.9mph
±0.0 mph vs preset
Hovered point radius
hover a chart
Most restrictive curvature at 86.2 m along the line — 50% of the way through the corner, right at the apex.
Curvature-limited speed · v_lim = √(μg / |κ|)
050100050100150 Distance (m) v_lim (mph)
Curvature |κ| along the line
0510152025050100150 Distance (m) |κ| (1/km)

Physical principle

A racing line is a free choice of path inside the track. Using more of the track width lets the line bend more gently, which raises the speed the tyres can sustain. But the apexThe point where the line comes closest to the inside of the corner. Here it is just a geometric control point; it has no built-in performance value. is only a geometric handle — moving it reshapes the curve; it does not add speed by itself.

Equations

κ = (x′y″ − y′x″) / (x′² + y′²)^(3/2)R = 1 / |κ|v_lim(s) = √(μ g / |κ(s)|)
κ
curvature at distance s (1/m)
R
local radius; ∞ on straights
s
distance along the line (m)

Your results

The line is 192.3 m long. Its tightest radius, 44.2 m, caps the speed at 48.9 mph with μ = 1.10.

Drag a control point to see how length and minimum radius trade against each other.

Try this

  • Move the apex later into the corner. Does the minimum radius improve? Does the total path length increase?
  • Switch between Early and Late Apex. Where along the line is the tightest point in each — before or after the apex?
  • Start from Late Apex and drag the apex (A) later and later, then compare the lines in Line Comparison. The time improves at first, then gets worse again: late is good, too late is not.
  • Pull the turn-in point toward the inside. How quickly does the minimum radius collapse?

The path is the car's centre, so the usable width (dashed) is the track width minus half the car width on each side. Curvature is computed analytically from the cubic Bézier segments; straights have exactly zero curvature. The line is C² through the apex but curvature jumps at turn-in and track-out (no clothoid transition) — real drivers smooth this.