Published on November 1, 2007
159.235 Graphics & Graphical Programming: 159.235 Graphics & Graphical Programming Lecture 24 - Projections - Part 1 Projections - Outline: Projections - Outline 3D Viewing Coordinate System & Transform Process Generalised Projections Taxonomy of Projections Perspective Projections 3D Viewing: 3D Viewing Inherently more complex than 2D case. Extra dimension to deal with Most display devices are only 2D Need to use a projection to transform 3D object or scene to 2D display device. Need to clip against a 3D view volume. Six planes. View volume probably truncated pyramid Coordinate Systems & Transform Process: Coordinate Systems & Transform Process Object coordinate systems. World coordinates. View Volume Screen coordinates. Raster Transform Project Clip Rasterize Generalised Projections.: Generalised Projections. Transforms points in a coordinate system of dimension n into points in one of less than n (ie 3D to 2D) The projection is defined by straight lines called projectors. Projectors emanate from a centre of projection, pass through every point in the object and intersect a projection surface to form the 2D projection. Projections.: Projections. In graphics we are generally only interested in planar projections – where the projection surface is a plane. Most cameras employ a planar film plane. But… the retina is not a plane - future devices such as direct retina devices may need more complex projections We will only deal with geometric projections – the projectors are straight lines. Many projections used in cartography are either non-geometric or non-planar. Exception – Image-based rendering - advanced topic Projections.: Projections. Henceforth refer to planar geometric projections as just: projections. Two classes of projections : Perspective. Parallel. A B A B A B A B Centre of Projection. Centre of Projection at infinity Parallel Perspective Parallel A Taxonomy of Projections: A Taxonomy of Projections Perspective Projections.: 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 Perspective: Perspective The first ever painting (Trinity with the Virgin, St. John and Donors) done in perspective by Masaccio, in 1427. Slide11: 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) Perspective Projections: Perspective Projections Lines parallel to a principal axis converge at an axis vanishing point. Perspective is categorized according to the number of such points. Corresponds to the number of axes cut by the projection plane. Perspective Projections: Perspective Projections z x y Projection plane x z y Lines parallel to a principal axis converge at an axis vanishing point. Categorized according to the number of such points Corresponds to the number of axes cut by the projection plane. 1-Point Projection: 1-Point Projection Projection plane cuts 1 axis only. 1-Point Perspective: 1-Point Perspective A painting (The Piazza of St. Mark, Venice) done by Canaletto in 1735-45 in one-point perspective 2-Point Perspective: 2-Point Perspective 2-Point Perspective: 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: 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. Intro to Projections -Summary: Intro to Projections -Summary 3D Viewing Coordinate System & Transform Process Generalised Projections Taxonomy of Projections Perspective Projections Clipping can be done in image space if more efficient – application dependent. Parallel Projections next… Acknowledgement - Thanks to Eric McKenzie, Edinburgh, from whose Graphics Course some of these slides were adapted. Parallel Projections: 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: 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 Axonometric Orthographic Projections: Axonometric Orthographic Projections Projection plane not normal to principal axis Show several faces of the object at once Foreshortening is uniform rather than being related to distance Parallelism of lines is preserved Angles are not Distances can be measured along each principal axis ( with scale factors ) Isometric Projection: 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. Slide24: Isometric Projection Normal Oblique projections.: 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: Oblique projection x z y Projection Plane Normal Parallel to x axis Geometry of Oblique Projections: Geometry of Oblique Projections Projection plane is x,y plane L=1/tan() - angle between normal and projection direction - Determines the type of projection is choice of horizontal angle. Given a desired L and , Direction of projection is (L.cos, L.sin,-1) Geometry of Oblique Projections: 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) and Mathematics of Viewing: Mathematics of Viewing Need to generate the transformation matrices for perspective and parallel projections. Should be 4x4 matrices to allow general concatenation. And there’s still 3D clipping and more viewing stuff to look at. Projections - Summary: Projections - Summary Orthographic matrix - replace (z) axis with point. Perspective matrix – multiply w by z. Clip in homogeneous coordinates. Preserve z for hidden surface calculations. Can find number of vanishing points. Acknowledgments - thanks to Eric McKenzie, Edinburgh, from whose Graphics Course some of these slides were adapted.