Planar Geometric Projection Classes

Projection A projection is a mapping of a three-dimensional (3D) space onto a two-dimensional subspace (i.e., a plane). The word projection also refers to the two-dimensional (2D) image resulting from such a mapping. Orientation of object with respect to projection plane

Classification of Planar Projections Comparison of projectors for perspective and parallel projection

PLANAR PARALLEL PROJECTIONS If the center of projection is infinitely far from the object, the projectors will be parallel to one another. ORTHOGRAPHIC PARALLEL Projection

ORTHOGRAPHIC PROJECTIONS Orthographic projection is a parallel projection technique, The parallel projectors are perpendicular (normal) to the projection plane

Object Position in Multiview Projection In a multiview projection, one object face and two principal axes are parallel to the projection plane. When projected, only one object face is visible. In terms of the orientation of the object with respect to the projection plane, multiview projection is identical to oblique projection, as well as to one-point perspective projection.

Orientation of object with respect to projection plane

Orthographic Projection Categories In an axonometric projection, all three principal axes are inclined to the projection plane. No axis is parallel (or perpendicular) to the projection plane. When projected, three principal faces of the object are visible. Axonometric projection results in an orthographic pictorial view. Object position in axonometric projection

Projection Classes with Orientation Shown

Projection Plane Location in a Parallel Projection

Projection Plane Location in a Perspective Projection

Projection Plane Location in a Projection

Perspective Projection In a perspective projection, the projectors converge to a single viewpoint called the center of projection (CP). The center of projection represents the position of the observer of the scene and is positioned at a finite distance from the object. Dürer's Artist Drawing a Lute illustrates perspective projection, with the eyebolt mounted on the wall serving as the center of projection.

Geometric Arrangement of an Oblique Projection Parallel projectors intersect the projection plane at an oblique angle. One principal face of the object is parallel to the projection plane. The object face that is parallel to the projection plane is projected true size.

Oblique Projection Two angles can be used to describe the intersection of an oblique projector with a projection plane An in-plane angle β measures the angle of rotation of the projector about the projection plane normal. The out-of-plane angle α is called the oblique projection angle. The oblique projection angle determines the type of oblique projection: cavalier, cabinet, or general.

Classes of Oblique Projections Cavalier Oblique Angle (α) used for the cube on the left is 45 degrees the measure of all of the edges of both the cube and its cavalier projection are identical. The receding (depth) axis is not foreshortened; it is scaled the same as the other (horizontal, vertical) principal axes. Cabinet Oblique Oblique projection angle (α) is approximately 63.43 degrees. The projected receding (depth) edge length is one-half that of the other (horizontal, vertical) edge lengths. The receding axis of a cabinet oblique is foreshortened to exactly one-half that of the other principal axes. projection that results from a cabinet oblique appears to the eye to be more proportionally correct than the cavalier oblique. This is because, visually, we expect some foreshortening to occur along a receding axis.

Oblique projection angle in 2D By collapsing the 3D oblique projection geometry into 2D, it is easy to understand why a 45 degree projection angle results in no foreshortening while a ~63.43 degree angle scales a projected length to one-half of the original. Figure shows a projection plane on edge and the object, a line of length L. The line is perpendicular to the projection plane The cube edges (not parallel to the projection plane) are perpendicular to the projection plane. Tan 45° = 1,

Depth (receding) axis In an oblique drawing, one axis is horizontal and another is vertical. The third, receding (or depth) axis can be inclined at any angle, but it is normally chosen to be 30, 45, or 60 degrees. This angle determines the relative emphasis of the receding planes on the projection,

Relationship between the receding axis angle and in-plane projector angle β The receding axis angle is related to the inplane projector angle β The receding axis angle is equal to β – 180 degrees, β is the in-plane angle of rotation of the oblique projector about the projection plane normal.

AXONOMETRIC PROJECTIONS Axonometric projections are classified according to the angles made by the principal axes when projected onto the projection plane. In a trimetric projection none of the angles between the projected principal axes are equal. In a dimetric projection, two of the three angles are equal. In an isometric projection, all three angles are equal. In the trimetric projection all three axes are foreshortened by different amounts, and all three projected angles are different. Two of the three projected axes in the dimetric projection are foreshortened by the same amount, and the angles opposite these axes are also equal

Isometric Projection Because all of the equal-length axes are inclined to the projection plane, all of their projections are foreshortened The amount of foreshortening is related to the projected angle. There is an equal amount of foreshortening along all three axes, and all of the angles are equal.

Perspective Projection Characteristics The center of projection is a finite distance from the object. The projectors are therefore nonparallel rays that converge to the center of projection. when parallel object edges are not parallel to the projection plane, the edges converge to a vanishing point when projected. objects or features that are farther away from the projection plane are more foreshortened (i.e., smaller) than closer ones. The principal advantage of perspective projection is that it produces a more realistic image. It closely approximates the view as seen by the human eye. Classes of Perspective Projection One-point perspective (one principal vanishing point) Two-point perspective (two principal vanishing points) Three-point perspective (three principal vanishing points)

Classes of Perspective Projection

Vanishing Points Inclined to the projection plane, then the projected edges will not be parallel; they will converge to a vanishing point. Both principal vanishing points lie on the horizon line. Each of these vanishing points results from the projection of a set of parallel object edges. Vanishing points for edges parallel to a plane always lie along a straight line in the projection plane, with the line parallel to the plane.

In a one-point perspective projection, one object face is parallel to the projection plane. One principal axis is perpendicular to the projection plane, and the other two principal axes (horizontal, vertical) are parallel to the projection plane.