First, let's validate this new setup by repeating the perspective problem introduced at the beginning: constructing the perspective image of a metric grid. This is done in two parts:

(1) The horizontal spacing of the orthogonals of the grid is found from the intersection of the orthogonals with the ground line; the intersections are projected into the perspective image by connecting them with their vanishing point (the principal point, vp).

(2) The vertical spacing of the transversals of the image grid is found by drawing a visual ray from the grid point in the ground plane to the viewpoint; the transversal is at the intersection of this visual ray with the orthogonal of the grid line in which the point lies.

Thus, two lines are necessary to locate an image point: (1) a visual ray from the physical point to the viewpoint, and, in central perspective, (2) anorthogonal from the ground line to the orthogonal vanishing point (which is the principal point in central perspective).

finding transversals with the visual ray method

In the diagram, the points a through e mark intersections in the metric grid on the ground plane. Points a, c and e line on the orthogonal that intersects the ground line at v; points b and d lie on the orthogonal that intersects the ground line at y.

The first step is to connect the ground line intersections (v to z) with the orthogonal vanishing point (vp), to create five orthogonals in the perspective image.

The second step is to connect each point by a visual ray (blue lines) to the viewpoint.

Each visual ray intersects its corresponding orthogonal at the image of the grid intersection in the image plane: this defines points a' to e'. Horizontal lines through these points define the transversals, or the metric grid units of perspective recession. (Compare this diagram with the earlier diagram based on the elevation and plan.)

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