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Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Lecture outline COMP30019 Graphics and Interaction Introduction to perspective geometry Perspective Geometry Perspective Geometry Adrian Pearce Centre of projection Department of Computer Science and Software Engineering University of Melbourne Projection using vectors The University of Melbourne Human perspective Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Perspective geometry Geometry of image formation Mapping from 3D space to 2D image surface, more specifically, a mapping from 3D directions (rays to/ from the observer). How are three-dimensional objects projected onto two-dimensional images? ◮ You can think perspective as a transformation as a way of moving from a higher dimensional image to a lower Aim: understand point-of-view, projective geometry. dimensional form. ◮ The X , Y , Z points in the three dimensional world, Reading: sometimes called voxels, are transformed in to x , y pixels ◮ Foley Sections 6.1 to 6.4 (excluding example 6.1, we’ll in a two-dimensional image. cover matrices later). Simplest device that does this is the pin-hole camera that gives Additional reading: perspective projection . ◮ Perspective is also covered in Chapter 3 of the Red Book. Practical cameras with lenses ideally give the same projection, aside from greater light gathering, and issues like focus. Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Pinhole Camera Perspective geometry projection screen (X,Y,Z) light ray from object for image (maybe translucent f X waxed paper) O image of object x Z (upside down) pinhole in box object in 3D scene light-tight box Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Perspective geometry Perspective Formulas Point P = ( X , Y , Z ) in 3D space has projection ( x , y ) in the image where ◮ Basically an abstraction of pin-hole camera. x X = f Z ◮ Look at XOZ plane—same thing happens in YOZ plane. y Y ◮ Actual point in 3D space is ( X , Y , Z ) = f Z ◮ 0 is origin (focal point) or centre of projection. or ◮ Z is distance from actual point to origin. Xf ◮ f is focal distance (focal length). = x Z ◮ x is the image (upside down) with respect to real world. Yf y = Z f being the “focal distance” (sometimes f is called d ). Look at similar triangles in the previous diagram. Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Perspective Formulas Alternative geometry (X,Y,Z) ◮ Look at perspective projection diagram to convince yourself of this — triangles xOf and XOZ have the same f X proportions. x ◮ Rearranging gives equations shown below. ◮ These formulas apply only for this special coordinate O system, sometimes called camera-centred coordinates, for which perspective projection has a particularly simple form. Z ◮ For other coordinate systems, some 3D transformation will be necessary (see later). Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Alternative geometry Centre of projection A A A' Projectors A' Projectors ◮ Image projection surface imagined to be in front of B B projection centre. B' B' Projection Projection ◮ Geometrically equivalent plane plane Center of ◮ Often more convenient. Center of projection projection at infinity (a) (b) Foley, Figure 6.03 Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective “One-point” perspective One point perspective projection (Foley, Figure 6.04) z -axis vanishing point y y z -axis vanishing point x x z z Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective One-point perspective projection (Foley, Figure 6.05) “Two-point” perspective y Projection plane x Center of projection z Projection plane normal Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective “Three-point” perspective Vanishing points ◮ In 3D, parallel lines meet only at infinity, so the vanishing point can be thought of as the projection of a point at infinity. ◮ If the set of lines is parallel to one of the three principal axes, the vanishing point is called an axis vanishing point . ◮ So called “one-point”, “two-point”, and “three-point” perspectives are just special cases of perspective projection, depending on how image plane lines up with significant planes in scene. ◮ Talking about these cases specifically is mainly an artifact of artists or architects dealing with horizontals and verticals in built environments. ◮ In fact, there are an infinity of vanishing points, one for each of the infinity of directions in which a line can be oriented. Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective Introduction to perspective geometry Perspective Geometry Centre of projection Projection using vectors Human perspective House example (Foley Section 6.4) One-point, centred perspective projection example y y x v x (8, 16, 30) (16, 10, 30) VUP CW VRP (16, 0, 30) u VPN (0, 10, 54) Window on n view plane DOP (16, 0, 54) z PRP = (8, 6, 30) z Foley Figures 6.21 and 6.22 Adrian Pearce University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry COMP30019 Graphics and InteractionPerspective Geometry