Graphics and Interaction Perspective Geometry
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1 Graphics and Interaction Perspective Geometr Department of Computer Science and Software Engineering The Lecture outline Introduction to perspective geometr Perspective Geometr Centre of projection Projection using vectors Human perspective
2 Perspective geometr How are three-dimensional objects projected onto two-dimensional images? Aim: understand point-of-view, projective geometr. Reading: Fole Sections 6.1 to 6.4 (ecluding eample 6.1, we ll cover matrices later). Additional reading: Perspective is also covered in Chapter 3 of the Red Book. Geometr of image formation Mapping from 3D space to 2D image surface, more specificall, a mapping from 3D directions (ras to/ from the observer). You can think perspective as a transformation as a wa of moving from a higher dimensional image to a lower dimensional form. The X, Y, Z points in the three dimensional world, sometimes called voels, are transformed in to, piels in a two-dimensional image. Simplest device that does this is the pin-hole camera that gives perspective projection. Practical cameras with lenses ideall give the same projection, aside from greater light gathering, and issues like focus.
3 Pinhole Camera projection screen for image (mabe translucent waed paper) light ra from object image of object (upside down) pinhole in bo object in 3D scene light-tight bo Perspective geometr (X,Y,Z) f X O Z
4 Perspective geometr Basicall 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). is the image (upside down) with respect to real world. Perspective Formulas Point P =(X, Y, Z ) in 3D space has projection (, ) in the image where or f f = X Z = Y Z = Xf Z = Yf Z f being the focal distance (sometimes f is called d). Look at similar triangles in the previous diagram.
5 Perspective Formulas Look at perspective projection diagram to convince ourself of this triangles Of and XOZ have the same proportions. Rearranging gives equations shown below. These formulas appl onl for this special coordinate sstem, sometimes called camera-centred coordinates, for which perspective projection has a particularl simple form. For other coordinate sstems, some 3D transformation will be necessar (see later). Alternative geometr (X,Y,Z) f X O Z
6 Alternative geometr Image projection surface imagined to be in front of projection centre. Geometricall equivalent Often more convenient. Centre of projection A A Projectors A' B Projectors A' B Center of projection Figure 6.03 (a) B' Projection plane Center of projection at infinit (b) B' Projection plane Fole,
7 One-point perspective One point perspective projection (Fole, Figure 6.04) z-ais vanishing point z-ais vanishing point z z
8 One-point perspective projection (Fole, Figure 6.05) Projection plane Center of projection z Projection plane normal Two-point perspective
9 Three-point perspective Vanishing points In 3D, parallel lines meet onl at infinit, so the vanishing point can be thought of as the projection of a point at infinit. If the set of lines is parallel to one of the three principal aes, the vanishing point is called an ais 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 specificall is mainl an artifact of artists or architects dealing with horizontals and verticals in built environments. In fact, there are an infinit of vanishing points, one for each of the infinit of directions in which a line can be oriented.
10 House eample (Fole Section 6.4) (8, 16, 30) (0, 10, 54) z (16, 0, 54) (16, 10, 30) (16, 0, 30) One-point, centred perspective projection eample v VUP VRP VPN n DOP CW u Window on view plane Fole Figures 6.21 and 6.22 z PRP = (8, 6, 30)
11 Which of the below is the centre of project in Fole Figure 6.22? VRP (view reference point) PRP (projection reference point) VPN (view plane normal) DOP (direction of projection) VUP (view-up vector) Is the view plane inbetween the centre of projection and the house or behind the centre of projection? Two-point perspective projection eample In a two-point projection of a house, left, the viewplane (defined b the view plane normal, VPN), right, cuts the z and aes (Fole Figures 6.17 and 6.25). View plane v u z VPN
12 Parallel projection Parallel projection introduces no perspective distortion (centre of projection plane (focal point) is at infinit Along with its variants it is useful in engineering drawings, where measurements must be taken. oblique projection if view plane is not perpendicular to projection. Geometric project classes Subclasses of planar geometric projections (Fole Figure 6.10). Planar geometric projections Parallel Perspective Orthographic Oblique One-point Top (plan) Front elevation Side elevation Aonometric Cabinet Cavalier Other Two-point Three-point Isometric Other
13 Ortographic (parallel) projection Parallel projection (also known as orthographic projection) is given b = X = Y (That is, just drop Z coordinate.) Also can have normal perspective which is scaled parallel (orthographic) projection = sx = (f /Z )X = sy = (f /Z )Y That is, it s like perspective projection in which objects are squashed to some constant fictitious depth Z instead of being at their true depths. Oblique parallel projection using vectors P d p (, ) v v origin Fig 4.3 Rowe
14 Oblique parallel projection using vectors All points are projected parallel to the projection vector d onto the plane. Note the projection vector is not perpendicular to the plane, else it would be an orthogonal projection. Point p is projected onto point v, v. Derivation of oblique parallel projection The vector equation of the (projection) line is r = p + td The intersection of this line with the plane is at z = 0, therefore t = p z d z and b substitution we obtain ( v, v )={p p zd d z, p p zd d z }
15 Derivation of oblique parallel projection r and d are vector parameters. r =(,, z) is a vector pointing to a point on the plane along the dotted line. r = p + td is the vector equation of a line passing through a point. t is a scalar parameter. Provided that d z is non-zero, otherwise if it is zero the projection direction is parallel to the plane so it does not intersect. Perspective projection using vectors C c P p c Q p (, ) v v origin Fig 4.5 Rowe
16 Perspective projection using vectors Point C is the centre of projection. Points P and Q are projected onto the plane at the points shown. For point P, the projection direction is given b the vector P C (this is calculated for each point). Derivation of perspective projection The projection passing through C and P is r = c + t(p c) at z = 0 c z t = (p z c z ) and b substitution we obtain ( v, v )={c c z p c p z c z, c c z p c p z c z } further simplification gives ( v, v )={ c p z c z p p z c z, c p z c z p p z c z }
17 Perspective of the human ee Human ee effectivel uses a kind of spherical projection: Retina is curved, though projection centre (in lens) isn t at centre of the eeball (therefore not planar geometric projection). Doesn t eactl match perspective projection. Onl a problem for ver wide fields of view. Perspective is basicall the right projection for putting a 3D scene onto a flat surface for human viewing. Other projections are possible for special effects, e.g. fish-ee lens. Summar Perspective geometr is based loosel on the pin-hole camera model that maps 3D points onto a 2D image plane The image plane ma thought of either behind a focal point or in between a vanishing point and the object. Computer graphics largel concerns planar geometric projections, generall perspective projection and sometimes parallel projection for specific applications. One-point, two-point and three-point projection variants arise according to how man times the viewplane cuts the ais planes.
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