Projection

Definition “ Projection can be defined as a mapping of point P(x,y,z) onto its image P’(x’,y’,z’) in the projection plane, which constitute the display surface”. The mapping is determind by a projection line called the “Projector” that passes through P and intersect the view plane. The intersection point is P’. P(x,y,z) Projector projection plane P’(x’,y’,z’)

Taxonomy of Projection Projections Perspective (Converging Projections) One Point (one point principal vanishing point) Two Point (two point principal vanishing point) Three Point (threes point principal vanishing point) Parallel (Parallel Projectors) Orthographic (projectors perpendicular to the view plane) Multi-view (view plane parallel to principal planes) Axonometric (view plane not parallel to principal planes) Isometric Di-metric Trimetric Oblique (projectors not perpendicular to view plane) General Cavalier Cabinet Taxonomy of Projection

Perspective Projection Basic Principles: - “ The techniques of projection are generalizations of the principles used by artists in preparing perspective drawing of 3D objects and scenes. The eye of the artist is placed at the center of projection, and the canvas (view plane).”

Mathematical Description A Perspective projection is determined by prescribing a center of projection and a view plane. The view plane is determined by its view reference point Ro and view plane normal N. the object point P is located in world co-ordinates at (x,y,z). The problem is to determine the image point co-ordinates P’(x’,y’,z’).

View Plane Z N X Y View Plane P1(x,y,z) P’1(x’,y’,z’) C P2 Center of projection N View Plane Normal X Y

Perspective Characteristics/Anomalies Perspective Foreshortening:- “ The farther an object is from the center of projection, the smaller it appears.” Example: -Square A is larger in size than square B but at vanishing point in viewing plane they appears to be of same size. Y Viewing plane A B Vanishing point

2. Vanishing Points These points are formed by the intersection of lines parallel to one of the three principal axis. The number of principal vanishing points is determined by the number of principal axis intersected by the view plane.

3. View Confusion Objects behind the center of projection are projected upside down and backward onto the view plane. Y P1 P2 P3 center of projection P3’ P2’ X X O P1’ Z

4. Topological Distortion The points of the plane that is parallel to the view plane & also passes through the Centre of projection are projected to infinity by the perspective projection. When we join the point which is back of the viewer to the point which is front of the viewer then the line will be projected as a broken line of infinite extent. Y A’ at infinity Centre of projection A Viewing Plane C B’ C’ A’ B INFINITY X

Types of Vanishing Points One Principal Vanishing Point Perspective Projection Two Principal Vanishing Point Perspective Projection Three Principal Vanishing Point Perspective Projection

One Principal Vanishing Point Perspective Projection This Perspective projection occurs when the projection plane is perpendicular to one of the Principal axis (x or y or z). Assume that it is z-axis. The view plane normal vector N is the vector N1 an Principal Vanishing Point is Therefore, V=V1I+V2J+V3K X=V1t+l Y=V2t+m Z=V3t +n P(l,m,n) K=N.V=n1v1+n2v2+n3v3 X3 =a1 Y3 =b1 Z3 =c1+ d1 n3 P3 : Where a1,b1,c1 are coordinates of the centre of projection n1,n2,n3 are components of view plane normal vector d1 is proportional to distance D from the view plane to the Centre of Projection

Two Principal Vanishing Point Perspective Projection “This Perspective projection occurs when the projection plane intersects exactly two of the principal axes.” VP2 VP1 Horizon Line

Continue……….. In this case where the projection plane intersects the x and y axes, for example, the normal vector satisfies the relationship N.K=0 or n3=0, and so the principal vanishing points are X1 =a1+ d1 n1 Y1 =b1 Z1 =c1 X2 =a1 Y2=b1+ d1 n2 Z2 =c1 P1 : P2 :

Three Principal Vanishing Point Perspective Projection This Perspective projection occurs when the projection plane intersects all the three principal axes x, y and z- axes. P2 P1 X1 =a1+ d1 n1 Y1 =b1 Z1 =c1 P1 X2 =a1 Y2 =b1+ d1 n2 Z2 =c1 P2 X3 =a1 Y3 =b1 Z3 =c1+ d1 n3 P3 P3

Parallel Projection Basic Principles: - “ The parallel projection used by drafters and engineers to create working drawings of an object which preserves its scale and shape. The complete representation of these details often requires two or more views (projections) of the object onto different view planes.

Continue………. In parallel projection, image points are found as the intersection of the view plane with a projector drawn from the object point and having a fixed direction. Direction of V Y Projection View Plane P(x,y,z) P’(x’,y’,z’) P2 P2’ N X Z

Mathematical Description A Parallel projection is determined by prescribing a direction of projection vector V and a view plane. The view plane is determined by its view reference point Ro and view plane normal N. the object point P is located in world co-ordinates at (x,y,z). The problem is to determine the image point co-ordinates P’(x’,y’,z’).

Types of Parallel Projection Orthographic Projection Oblique Projection

Orthographic Projection “Projections are characterized by the fact that the direction of projection is perpendicular to the viewing plane. They are used to produce the front, side and top views of an object.” Example: - Engineering & architectural drawings employ it.

Categories/ Types of Orthographic Projections Multiview Projections: - “When the direction of projection is parallel to any of the principal axis, this produces front, top and side views of an object.” Example: -mechanical drawings employ it.

Continue………… Axonometric Projections: - “These projections are those in which the direciton of projection is not parallel to any of the three principal axis.” Some common sub-categories of Axonometric Projections are: - Isometric Di-metric Tri-metric

Continue……………. Isometric Projection: -The direction of projection makes equal angles with all the three principal axes. Di-Metric Projection: - The direction of projection makes equal angles with exactly two of the principal axes. Tri-Metric: - The direction of projection makes unequal angles with the three principal axes.

Oblique Projection “Projection obtained by projecting points along parallel lines that are not perpendicular to viewing plane i.e. at any angle of consideration is called oblique parallel projections.” OR “Non-orthographic parallel projections are Called oblique parallel projections.”

SUB-CATEGORIES OF AXONOMETRIC PROJECTIONS ARE: - Cavalier: - “ The direction of projections is chosen so that there is no fore-shortening of lines perpendicular to xy-plane.” Cabinet: - “The direction of projections is chosen so that lines perpendicular to xy-plane are fore-shortening by half their lengths.”

Cabinet Projection y C’ D’ E’ H’ x A’ G’ B’