Methods of Projections Parallel and Central Projection

Home , Axonometry, Isometric projection, Oblique projection, Parallel projection, Projection plane

Methods of Projections Parallel and central projection

Learning Objectives • Introduce the classical views • Compare and contrast image formation by computer with how images have been formed by architects, artists, and engineers • Learn the benefits and drawbacks of each type of view • Introduce the mathematics of projection

Linear mapping

1. Image of a point is point 2. Image of a line is line (or point) 3. Image of a plane is plane (or line)

1 View

We call the pictures for observing objects with different ways are views. View results are related with the shape and size of scene objects, the position and direction of viewpoint.

Planar Geometric Projections

• Standard projections project onto a plane • Projectors are lines that either – converge at a center of projection – are parallel • Such projections preserve lines – but not necessarily angles • Nonplanar projections are needed for applications such as map construction

2 Classical Projections

Perspective vs Parallel

• Computer graphics treats all projections the same and implements them with a matrix • Classical viewing developed different techniques for drawing each type of projection • Fundamental distinction is between parallel and perspective viewing even though mathematically parallel viewing is the limit of perspective viewing

3 Multiview Orthographic Projection

• Projection plane parallel to principal face • Usually form front, top, side views isometric (not multiview orthographic view) front in CAD and architecture, we often display three multiviews plus isometric

side top

Advantages and Disadvantages

+ Preserves both distances and angles Shapes preserved Can be used for measurements • Building plans • Manuals - Cannot see what object really looks like because many surfaces hidden from view Often we add the isometric

4 Types of Axonometric Projections

Isometric projection

Isometric projection- a drafting style in which a two-dimensional object is drawn to look three- dimensional. The horizontal edges of the object are drawn at a 30° or 150° angle, and all vertical lines at 90° (remain vertical).

5 Advantages and Disadvantages

• Lines are scaled (foreshortened) scaling factors are known only for a axis (principal directions) Projection of a circle in a plane not parallel to the projection plane is an ellipse • Does not look real for large objects because far objects are scaled the same as near objects • Used in CAD applications (with perspective) 13

Axonometry

6 M. C. Escher, Waterfall, 1961

Taska: Tangent plane of the cone Regular hexagon Helix Hyperboloid of revolution

Oblique Projection

Arbitrary relationship between projectors and projection plane

7 Advantages and Disadvantages

• Can pick the angles to emphasize a particular face – Architecture: plan oblique, elevation oblique • Angles in faces parallel to projection plane are preserved while we can still see “around” side • In physical world, cannot create with simple camera;

Orthographic

and Oblique drawings

8 Pohlke’s Theorem (1860)

The principal theorem of oblique axonometry

Three straight line segments of arbitrary length in a plane, drawn from a point under arbitrary angles form a parallel projection of three equal segments drawn from the origin on three perpendicular coordinate axes. However, only one of the segments or one of the angles may vanish.

Any complete plane quadrangle can serve as the parallel projection of a tetrahedron similar to a given one.

https://www.geogebra.org/m/CsamWsfy

Sketching

Method of pictorial sketching of an object

1. Sketch an enclosing box of the given object. If the object decomposes from different parts we sketch the boxes circumscribed to these parts. 2. Sketch views in the principal faces (thin lines). 3. Sketch auxiliary parallel lines 4. Thick lines to highlight visible edges.

9 Sketching-isometric grid

The horizontal edges of the object are drawn at a 30° or 150° angle, and all vertical lines at 90° (remain vertical).

Perspective Projection

Projectors converge at center of projection

Naturally we see things in perspective o Objects appear smaller the farther away they are; o Rays from view point are not parallel. 22

10 Vanishing Points • Parallel lines (not parallel to the projection plan) on the object converge at a single point in the projection (the vanishing point) • Drawing simple perspectives by hand uses these vanishing point(s)

11 Types of Perspective

Three-Point Two-Point One-Point

• Number of principal face parallel to projection plane • Number of vanishing points for cube

Advantages and Disadvantages

• Objects further from viewer are projected smaller than the same sized objects closer to the viewer (diminution) – Looks realistic • Equal distances along a line are not projected into equal distances (nonuniform foreshortening) • Angles preserved only in planes parallel to the projection plane • More difficult to construct by hand than parallel projections (but not more difficult by computer)

12 Yp Window Image coordinate plane

3D space Window Viewing Volume Projection Xp

2D image Far plane

World coordinate system Screen coordinate system

Z Near plane

Viewpoint

Viewport 27

Matrix Reprezentation of Projection

Orthogonal projection to coordinate plane p = (xy)

xxp 

yyp 

z p  0 3 X(,,)(,,) x y z  E  X  xp y p z p p 1 0 0   X PX; P  0 1 0 000

13 Simple Perspective Consider a simple perspective with the COP at the origin, projection (picture) plane p parallel to coord. plane (x,y).

x d X(,,) x y z p  xz y d X  (,,) x y z p  p p p yz

zdp 

Matrix reprezentation only in the projective space!

Perspective Matrices Simple projection matrix in homogeneous coordinates d xx  d d   d d  p z X  x, y , d   x , y , d ,1   xdydzdz , , ,  z z   z z  d yy p z d 0 0 0  zdp  0d 0 0 X'; PX P  0 0d 0  0 0 1 0