3. The Pinhole Camera

A camera is sometimes treated as a box of familiar settings: position, target and field of view. Enter the numbers, call a library function and a view appears. That convenience is useful after the camera model is understood. Here it would hide the main construction. A ray tracer needs one direction for every image sample, and the camera is the rule that produces it.

We shall use a pinhole camera. All primary rays begin at one origin and pass through chosen positions on a conceptual viewport. There is no lens area and therefore no depth of field. The model is limited, but it isolates the relationship we need: a rectangular sample position becomes a direction in the camera's three-dimensional coordinate frame.

Construct the camera basis

Choose an origin C, a target point T and a reference direction called worldUp. The camera forward direction is the normalised displacement from origin to target. The right direction is perpendicular to forward and world up. A corrected up direction is then perpendicular to right and forward.

const forward = normalise(subtract(target, origin));
const right = normalise(cross(forward, worldUp));
const up = cross(right, forward);

Why construct up again instead of using worldUp directly? The reference vector expresses which way we would prefer the camera to roll. Unless the camera looks exactly level, it need not be perpendicular to forward. The corrected vector completes an orthonormal basis. If the three directions are unit length and pairwise perpendicular, each viewport displacement can be assembled without changing scale unpredictably.

Turn field of view into viewport size

The vertical field of view is an angle. Place the viewport one arbitrary unit in front of the camera. Half of its height and that unit distance form a right triangle, so the half-height is the tangent of half the angle. The horizontal half-width is then scaled by the raster aspect ratio.

halfHeight = tan(verticalFov / 2)

halfWidth = aspectRatio × halfHeight

A wider field of view increases these extents. The same number of pixels then spans a wider cone of directions, so objects appear smaller. This is not the camera moving backwards. Both changes may place more of the scene in the image, but they do not produce the same perspective relationships.

Map one pixel centre

Let x and y identify a raster pixel. First map the centre to normalised image coordinates:

const u = (x + 0.5) / width;
const v = (y + 0.5) / height;

u increases from zero to one across the image. Browser v increases downwards, while camera up should increase upwards. We therefore use 1 - 2v for the vertical camera displacement and 2u - 1 for the horizontal displacement.

const horizontal = scale(right, (2 * u - 1) * halfWidth);
const vertical = scale(up, (1 - 2 * v) * halfHeight);
const direction = normalise(add(forward, add(horizontal, vertical)));

const ray = { origin, direction };

The centre sample of the centre pixel points close to forward. A top-right sample adds positive right and positive up. Every direction is normalised after those displacements are combined. We now have the ray equation from Lesson 2 and a repeatable way to supply its values for the whole raster.

Inspect one camera ray before constructing thousands

There is still no geometry to see, so a coloured scene would have no defined meaning. Instead, the laboratory keeps the image plane visible. Select one pixel, use its centre as the sample position and follow that one position through the camera calculation. That field-of-view value determines both the size of the image plane and the resulting direction.

Two diagrams describe one construction at different scales. A grid identifies which sample was chosen. The camera view shows the fixed origin, the image plane and the route through that sample. A numerical record then states the image-plane position and normalised direction. This proves that the selected pixel obtains a finite primary direction. It does not prove that the direction meets an object. Lesson 4 supplies that geometric question.

Recognise the invalid camera

If the origin and target are the same point, their displacement is zero and there is no forward direction. If forward is parallel or anti-parallel to worldUp, their cross product is zero and there is no unique right direction. These are not merely numerical edge cases. The supplied values do not define the requested frame.

A careful camera constructor validates both conditions. Quietly inserting a default direction may produce an image, but that image no longer represents the stated camera. Failure near the cause is more useful than a field of NaN values discovered after thousands of pixels have been processed.

Every sample can now ask a directed question from the camera. Lesson 4 gives that question a sphere equation and accepts only the pixels whose rays actually meet the surface.