Perpendicular Offset Calculator

A perpendicular offset is the shortest distance from an external point to a line. In surveying and civil engineering it is one of the most frequently computed quantities: every road cross-section, boundary stakeout, and pipeline alignment check reduces to dropping a perpendicular from a known point onto a design centreline.

The foot of perpendicular (also called the dropped foot or orthogonal projection) is the point on the line that is closest to the external point. Its position along the line is described by the t parameter — a value between 0 and 1 when the foot falls within the segment, and outside that range when the line must be extended to reach it.

The left/right side classification tells you which side of the line the point sits on, measured relative to the direction from A to B. Standing at A and facing B, a point to your left is a Left offset; a point to your right is a Right offset. This matches road-design convention where lane widths, drainage channels, and utility offsets are always referenced to the centreline direction.

This calculator uses a local flat-Earth projection to convert geographic coordinates into a metric Cartesian frame, then applies coordinate geometry to find the foot. The key steps are:

1. Project all three coordinates (A, B, P) to a local (easting, northing) frame in metres, using the mid-latitude cosine correction to keep east and north distances proportional.
2. Compute the scalar projection parameter: t = (P − A) · (B − A) / |B − A|². This is the normalised distance from A where the foot lands.
3. Compute the foot coordinates in the local frame: F = A + t × (B − A).
4. Convert F back to latitude and longitude.
5. Compute offset distance as the Euclidean distance from P to F in the local metric frame.
6. Determine the left/right side using the sign of the 2D cross product (B − A) × (P − A).

This approach is accurate to better than 0.1 % for line segments under 50 km, which covers the full range of construction staking and cadastral survey work.

The side classification uses the right-hand rule in a north-up coordinate system. Imagine standing at Line Start A and looking toward Line End B:

• Left — the point is to the left of the A→B direction (positive cross-product z-component).
• Right — the point is to the right (negative z-component).
• On Line — the cross product is effectively zero, meaning P already lies on the line.

In road design, the alignment direction defines the stationing direction (increasing chainages go from A to B). Left and right offsets in the results match what a survey crew would read from a design drawing — no conversion needed.

Construction staking (also called setting out or layout) converts a design plan into physical marks in the ground. Perpendicular offsets appear in three common scenarios:

• Centreline check: A known GPS point (a utility pole, fence corner, or survey monument) is checked against a proposed road centreline. The offset distance and left/right side tell the engineer how far off the existing feature is from the design.
• Offset stake placement: Construction stakes are driven at a fixed offset from the centreline (e.g. 5 m Right) so earthworks crews can work without disturbing the actual centreline marker. The foot of perpendicular gives the stake's position on the centreline.
• Pipe and cable routing: Utility corridors follow a design alignment. The perpendicular offset from each structure to the pipeline centreline confirms clearance requirements are met.

If the t parameter is less than 0, the foot of perpendicular falls before point A on the extended line. If t is greater than 1, it falls beyond point B. In either case, the nearest point on the segment is the closest endpoint, not the foot.

The calculator always reports the true geometric foot (which may lie on the extension), and flags the result with a yellow warning when the foot is outside the segment. This is important for construction staking: if the offset foot is beyond the end of the submitted alignment, the crew needs to extend the alignment or use a different reference line.