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Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective COMP30019 Graphics and Interaction Perspective Geometry Adrian Pearce Department of Computing and Information Systems University of Melbourne The University of Melbourne Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Lecture outline Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Perspective geometry How are three-dimensional objects projected onto two-dimensional images? Aim: understand point-of-view, projective geometry. Reading: ◮ Foley Sections 6.1 to 6.4 (excluding example 6.1, we’ll cover matrices later). ◮ Akenine-Moller Section 2.3. Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Viewing Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Viewport Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Geometry of image formation Mapping from 3D space to 2D image surface, more specifically, a mapping from 3D directions (rays to/ from the observer). ◮ You can think perspective as a transformation as a way of moving from a higher dimensional image to a lower dimensional form. ◮ The X , Y , Z points in the three dimensional world, sometimes called voxels, are transformed in to x , y pixels in a two-dimensional image. Simplest device that does this is the pin-hole camera that gives perspective projection . Practical cameras with lenses ideally give the same projection, aside from greater light gathering, and issues like focus. Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Pinhole Camera projection screen light ray from object for image (maybe translucent waxed paper) image of object (upside down) pinhole in box object in 3D scene light-tight box Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Perspective geometry (X,Y,Z) f X O x Z Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Perspective geometry ◮ Basically an abstraction of pin-hole camera. ◮ Look at XOZ plane (same thing happens in YOZ plane). ◮ Actual point in 3D space is ( X , Y , Z ) ◮ 0 is origin (focal point) or centre of projection. ◮ Z is distance from actual point to origin. ◮ f is focal distance (focal length). ◮ x is the image (upside down) with respect to real world. Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Virtual camera Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Virtual camera geometry (X,Y,Z) f X x O Z Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Virtual camera geometry ◮ Image projection surface imagined to be in front of projection centre. ◮ Geometrically equivalent ◮ Often more convenient to think about projection Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Perspective Formulas Point P = ( X , Y , Z ) in 3D space has projection ( x , y ) in the image where x X = f Z y Y = f Z or Xf x = Z 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 COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Perspective Formulas ◮ Look at perspective projection diagram to convince yourself of this — triangles xOf and XOZ have the same proportions. ◮ Rearranging gives equations shown below. ◮ These formulas apply only for this special coordinate system, sometimes called camera-centred coordinates, for which perspective projection has a particularly simple form. ◮ For other coordinate systems, some 3D transformation will be necessary (see later). Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Camera transformation Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective Centre of projection A A A' Projectors A' Projectors B B B' B' Projection Projection plane plane Center of Center of projection projection at infinity (a) (b) Foley, Figure 6.03 Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human 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 COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective One-point perspective projection (Foley, Figure 6.05) y Projection plane x Center of projection z Projection plane normal Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective “Two-point” perspective Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective “Three-point” perspective Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human 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 COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective House example (Foley Section 6.4) y x (8, 16, 30) (16, 10, 30) (16, 0, 30) (0, 10, 54) (16, 0, 54) z Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry

Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective One-point, centred perspective projection example y x v VUP CW VRP u VPN Window on n view plane DOP PRP = (8, 6, 30) z Foley Figures 6.21 and 6.22 Adrian Pearce University of Melbourne COMP30019 Graphics and InteractionPerspective Geometry