Pan a camera to point at a target: yaw angle from coordinates

You know where the target is — its forward and lateral offsets in your frame — and you need the one number that points the camera at it: the pan yaw angle. For a camera and a target sharing a horizontal plane, that number is a single atan2 away. This guide gives the formula with its sign conventions, a worked example with the arithmetic shown, the bench steps from offsets to a centred image, and the checks that catch the classic failures. If your offsets live in pixels or in another machine's coordinate frame, the linked calibration articles show how to bring them into millimetres first.

The formula and the conventions that make it work

For a pan-only camera and a target in the same horizontal plane, the yaw from the boresight to the target is:

Δx is the forward offset from the pan axis to the target, measured along the direction the camera faces at zero yaw. Δy is the lateral offset, positive to the right as the camera faces the scene. yaw is the angle to rotate the pan axis, positive to the right — clockwise seen from above. range is the straight-line distance to the target across the plane, which you reuse for focus and field-of-view checks.

Use the two-argument form, not arctan(Δy ÷ Δx): atan2 reads the signs of both inputs and returns −180° to +180°, so it stays correct when the target sits behind the camera (the atan2 definition spells out the quadrant behaviour). Units cancel in the ratio, so millimetres in give degrees out — mixing millimetres and inches between the two offsets is the only unit mistake available.

Worked example: target 200 mm ahead, 100 mm right

With Δx = 200 mm and Δy = 100 mm:

yaw = atan2(100 mm, 200 mm) = arctan(100 ÷ 200) = arctan(0.5) = 26.6° (0.4636 rad)
range = sqrt(200² + 100²) mm = sqrt(50,000) mm = 223.6 mm

Command 26.6° to the right and the target lands on the optical axis, 223.6 mm out along the new boresight.

Now put the target behind and to the right: Δx = −150 mm, Δy = 100 mm. A plain arctan(100 ÷ −150) = arctan(−0.667) returns −33.7°, which points ahead and to the left — the mirror position. atan2 sees the negative forward offset and returns 180° − 33.7° = 146.3°, the correct over-the-shoulder bearing. That 180° flip is the most common pointing error, and the two-argument form is the entire fix.

Keeping the target inside the field of view

A correct pan puts the target on the optical axis, so the field of view only needs to cover the target's surroundings plus your aim error. The width a lens covers at the target distance is:

At the worked range of 224 mm, a 40° horizontal angle of view covers 2 × 224 mm × tan 20° = 2 × 224 × 0.364 = about 163 mm. Your aim error competes against the half-angle, not the full field: ±1° of stage repeatability disappears inside a ±20° half-field, while the same ±1° with a 4° lens (±2° half-field) spends half the margin. Choose the field first; the calculator below turns that field and your sensor into a focal length.

What you need

Three things: a camera that rotates about a defined vertical axis, the two offsets in one frame, and something that evaluates atan2 — a script, a spreadsheet cell or a controller in the cell. The bench parts look like this:

The LS-MNT-0220 is the pan axis itself: a manual XY and rotation stage with a dia. 60 mm base, so you can mark zero, turn to the computed angle and trim the lateral aim without moving the camera.

Beneath it, the LS-MNT-0040 slide-table stand holds the assembly at a 700 mm height on a 400 × 450 mm base — enough bench room to lay the offsets out and measure them with a rule.

In an automated cell, the MV-VC2140-128G40-1T vision controller computes the angle and captures the verification frame on its N97 processor with 8 GB of memory, firing its four built-in light channels for the check image.

If the offsets live in pixels rather than on a drawing, run an image-scale calibration first — calibrating image scale automatically from a ruler in the scene walks through it. If they live in a robot or conveyor frame, transform them into the camera frame before the arctangent; the eye-to-hand setup for a camera above a robot base shows the chain.

Steps: from offsets to a centred image

1. Fix the frame. Origin on the pan axis, +x along the boresight at zero yaw, +y to the right, and write the convention on the stand. Every number after this inherits it.

2. Get Δx and Δy in one unit. Read them from a layout drawing, measure them on the bench, or convert them from pixels with an image-scale calibration. Transform first if they start in a robot or conveyor frame.
3. Compute yaw = atan2(Δy, Δx). Check the return unit of whatever evaluates it: if it returns radians, convert before commanding — 0.4636 rad × 57.2958 = 26.6°. An unconverted value commands the axis 0.46 "degrees" and the failure looks like a dead drive, not a maths bug.

4. Command the angle, positive to the right. On a manual stage, set the scale to zero and turn 26.6° right for the worked target.
5. Verify in the image. Capture a frame, find the target, convert the residual pixel offset to an angle. At 1,920 px across a 40° field, one pixel is 40 ÷ 1,920 = 0.0208°, so 48 px off centre is about 1.0° off boresight. Near the centre the linear ratio is close enough; farther out, use arctan(pixel offset ÷ focal length in pixels) instead. Add the residual to the commanded angle and re-shoot.
6. Log the residual per aim. A steady residual means a systematic offset, usually offsets measured from the camera body instead of the axis. A changing residual means play or slip in the drive.

Common mistakes

In the order they usually happen:

• Radians commanded as degrees — the factor is 57.3 and the axis barely moves.
• Sign flipped between your frame and the drive: the camera points the mirror direction and the miss is twice the computed yaw.
• arctan(Δy ÷ Δx) instead of atan2 — exactly 180° wrong for any target behind the axis.
• Offsets measured from the camera body rather than the pan axis; the axis is what rotates, so it is the origin the formula wants.
• Ignoring Δz: height does not change yaw, but a pan-only mount needs a tilt axis to centre a target above or below the plane.
• Evaluating atan2(0, 0) for a target dead ahead, where the function is undefined — test for Δx > 0 with Δy = 0 and command 0°.

Before you command the axis

• Δx and Δy carry the same unit; the ratio inside atan2 is dimensionless.
• The origin sits on the pan axis, not on the camera body.
• The sign convention is written where the mount can read it: right-positive.
• You know the evaluator's return range, −180° to +180°, and its unit.
• The axis can physically reach the angle — 146.3° needs clearance behind the camera.
• The image residual after correction is logged; that number is your real aim accuracy.

When a calibration beats pointing maths

Pointing maths answers "which way should it look". If the cell needs the camera to report where things are, repeatedly and in world coordinates, fix the mount and calibrate once instead — the transform chain from pixels to world coordinates is this same geometry done thoroughly, and the eye-in-hand pixel-to-world guide shows how the frames chain together. When an aimed pick still lands offset after the maths checks out, the pick point offset diagnosis separates mechanical offsets from calibration error. For the rest of the subject, the systems guides collect the calibration and frame topics.

Build the pan axis

Size the mount and the computer together: the camera stands category lists stands, slide tables and rotary stages, the camera stand finder filters them by type, and the stand configurator walks mount, camera and light in order. The vision controllers category covers the computing side for an automated cell. If you want the offsets and the computed angle checked against a specific mount before ordering, send the layout through the contact page.