Gobo Projected Image Size and Keystone

ScopeA 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, more than the narrow-beam 1/cos rule because the far edge of the beam lands flatter than the centre, while the perpendicular axis stays near its straight-on width, which is what makes a projected logo look like a trapezoid: for a 36-degree field the long axis is 1.58 times the straight-on diameter at 45 degrees and 2.93 times at 60. 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 30.8 ft while illuminance falls to 0.71, half a stop down. A logo that reads 19.5 ft wide straight on becomes 30.8 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.

Formula or tableimage_diameter_ft = 2 x throw_ft x tan(phi), phi = field_angle_deg / 2; stretched_axis_ft = throw_ft x sin(phi) x [1/cos(incidence - phi) + 1/cos(incidence + phi)] (the two edge rays meeting the tilted surface); keystone_stretch = stretched_axis_ft / image_diameter_ft; relative_illuminance = cos(incidence_deg) at the beam centre; no finite image once incidence + phi reaches 90 degrees.

Source and editionGobo image size from the field-angle cone, with the long axis from the edge rays meeting the tilted surface and the 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.

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

What governsEstimate. AHJ and licensed professional govern.

API field namesthrow_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 is throw x sin(phi) x [1/cos(incidence - phi) + 1/cos(incidence + phi)], more than the narrow-beam 1/cos(incidence); the perpendicular axis stays near its straight-on widthprojection geometry
• Illuminance falls by cos(incidence) as the same light covers more areaphotometry