Gobo Projected Image Size and Keystone

A gobo fills the fixture's field, so the projected image is the field-angle cone intersected with the surface: straight on, a circle whose diameter is twice the throw times the tangent of half the field angle, the same geometry as the beam pool applied to the image rather than the light. Off perpendicular that circle becomes an ellipse. The axis in the plane of the tilt stretches by one over the cosine of the incidence angle while the perpendicular axis does not stretch at all, which is what makes a projected logo look like a trapezoid: at 45 degrees the stretch is 1.41, and at 60 degrees it is 2.00, so the image is twice as long as it is wide. The same cosine works against you on brightness, because the light is spread over more area and illuminance falls by the cosine of the same angle. A 36-degree ellipsoidal at a 30 ft throw makes a 19.5 ft circle straight on; hang it 45 degrees off perpendicular onto a back wall, an ordinary front-of-house angle, and the long axis goes to 27.6 ft while illuminance falls to 0.71, half a stop down. A logo that reads 19.5 ft wide straight on becomes 27.6 ft tall and noticeably dimmer, and it needs optical keystone correction, a distorted gobo cut to compensate, or a better hanging position -- and the case for the better position is worth making before the gobo is ordered. When the gobo's usable image is smaller than the gate, the projection scales by that fraction. A geometric estimate; the fixture's published field angle and a focus check in the room govern.

image_diameter_ft = 2 x throw_ft x tan(field_angle_deg / 2); keystone_stretch = 1 / cos(incidence_deg); stretched_axis_ft = image_diameter_ft x keystone_stretch; relative_illuminance = cos(incidence_deg).

Gobo image size from the field-angle cone with the 1/cos keystone stretch and cos illuminance falloff for a non-perpendicular hit, by name; public projection geometry. The fixture's published field angle and a focus check in the room govern.

The projection geometry is public. The throw, field angle, incidence angle, and gobo dimensions are the designer's own values.

Estimate. AHJ and licensed professional govern.

Field names used by the API: throw_ft, field_angle_deg, incidence_deg, gobo_image_mm, gate_diameter_mm, image_diameter_ft, keystone_stretch, stretched_axis_ft, relative_illuminance

• Field angle the image fills the field, so diameter = 2 x throw x tan(field/2)projection geometry
• Keystone the tilt-plane axis stretches by 1/cos(incidence); the perpendicular axis does not stretchprojection geometry
• Illuminance falls by cos(incidence) as the same light covers more areaphotometry