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Computer Graphics Projections 3D Viewing • Inherently more complex than 2D case. – Most display devices are only 2D • Need to use a projection to transform 3D object or scene to 2D display device. 2 Generalised Projections. • Transforms points in a coordinate system of dimension n into points in one of less than n (ie 3D to 2D) • Projectors emanate from a centre of projection, pass through every point in the object and intersect a projection surface to form the 2D projection. 3 Projections. • Henceforth refer to planar geometric projections as just: projections. • Two classes of projections : – Perspective. – Parallel. Parallel A A Parallel A A Centre of Projection. B B B Centre of B Projection Perspective at infinity A Taxonomy of Projections Planar geometric projections. Parallel Perspective Orthographic Oblique 1 point Axonometric Cabinet Cavalier 2 point Isometric 3 point Elevations Perspective Projections. • Defined by projection plane and centre of projection. • Visual effect is termed perspective foreshortening. – The size of the projection of an object varies inversely with distance from the centre of projection. – Similar to a camera - Looks realistic ! • Not useful for metric information – Parallel lines do not in general project as parallel. – Angles only preserved on faces parallel to the projection plane. – Distances not preserved Graphics 6 Perspective Projections • A set of lines not parallel to the projection plane converge at a vanishing point. – Can be thought of in 3D as the projection of a point at infinity. – Homogeneous coordinate is 0 (x,y,0) 7 1-Point Projection Projection plane cuts 1 axis only. 1-Point Perspective A painting (The Piazza of St. Mark, Venice) done by Canaletto in 1735- 45 in one-point perspective 2-Point Perspective y z x Projection plane 2-Point Perspective Painting in two point perspective by Edward Hopper The Mansard Roof 1923 (240 Kb); Watercolor on paper, 13 3/4 x 19 inches; The Brooklyn Museum, New York 3-Point Perspective Generally held to add little beyond 2-point perspective. A painting (City Night, 1926) by Georgia O'Keefe, that is approximately in three-point perspective. y z x Projection plane Parallel Projections • Specified by a direction to the centre of projection, rather than a point. – Centre of projection at infinity. • Orthographic – The normal to the projection plane is the same as the direction to the centre of projection. • Oblique – Directions are different. Orthographic Projections Most common orthographic Projection : Front-elevation, Side-elevation, Plan-elevation. Angle of projection parallel to principal axis; projection plane is perpendicular to axis. Commonly used in technical drawings 15 Isometric Projection • Most common axonometric projection • Projection plane normal makes equal angles with each axis. • i.e normal is (dx,dy,dz), |dx| = |dy|=|dz| • Only 8 directions that satisfy this condition. Isometric Projection y y Normal 120º 120º 120º x z x Projection All 3 axes equally foreshortened Plane z - measurements can be made - Hence the name iso-metric Oblique projections. • Projection plane normal differs from the direction of projection. • Usually the projection plane is normal to a principal axis. – Projection of a face parallel to this plane allows measurement of angles and distance. – Other faces can measure distance, but not angles. – Frequently used in textbooks : easy to draw ! Oblique projection Normal Parallel to x axis y x Projection Plane z Geometry of Oblique Projections • Point P=(0,0,1) maps to: P’=(l.cosa, l.sina, 0) on xy plane, and P(x,y,z) onto P’(xp,yp,0) 1 0 l cosa 0 x p x z (l cosa ) 0 1 l sin a 0 and M ob y p y z (l sin a ) 0 0 0 0 0 0 0 1 Perspective Projection – Simplest Case Centre of projection at the origin, Projection plane at z=d. Projection Plane. y P(x,y,z) x Pp(xp,yp,d) d z Perspective Projection – Simplest Case From similar triangles : xp x yp y x ; d z d z xp P(x,y,z) dx x dy y xp ; yp z z z/d z z/d d y d P(x,y,z) z x Pp(xp,yp,d) yp P(x,y,z) d z y Perspective Projection x T d 1 d .x d.y x y z z T T yp d 1 p z z d The transform d ation can be represente as a 4x4 matrix : 1 0 0 0 0 1 0 0 M per 0 0 1 0 0 0 1/d 0 Perspective Projection Represent the general projected point Pp X Z W T Y 1 0 0 0 x 0 1 0 0 y Pp M per P 0 0 1 0 z 0 0 1/d 0 1 X Z W x z / d T T Y y z 24 Orthographic Projection Orthographic Projection onto a plane at z = 0. xp = x , yp = y , z = 0. 1 0 0 0 0 1 0 0 M orth 0 0 0 0 0 0 0 1

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posted: | 3/14/2012 |

language: | English |

pages: | 24 |

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graphics lecture

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