1 Classic perspective – properties and drawbacks Fast generation of curved perspectives for ultra-wide-angle lenses in VR applications Georg Glaeser 1 , Eduard Gröller 2 1 Geometry Department, University of Applied Arts in Vienna, Oskar-Kokoschka-Platz 2, A-1010 Vienna, Austria e-mail: georg.glaeser@uni-ak.ac.at, http://www.uni-ak.ac.at/geom/glaeser.html 2 Institute of Computer Graphics, Vienna University of Technology, Karlsplatz 13/186/2, A-1040 Vienna, Austria e-mail: groeller@cg.tuwien.ac.at, http://www.cg.tuwien.ac.at/home Classic ultra-wide-angle perspectives are not realistic and often misleading, neverthe- less, they have to be used in many appli- cations where the viewer needs a survey of the scene. Current hardware, however, only supports classic perspectives. We present a fast polygon-oriented algorithm that al- lows the use of curved perspectives in order to overcome several drawbacks, but at the cost of non-linearity. Space is projected onto a sphere rather than onto an image plane. The spherical coordinates are then interpreted two-dimensionally. We discuss the advan- tages and drawbacks of several approaches to curved perspectives. Time measurements show that real-time animation of reason- ably complicated scenes can still be done; the overhead (additional cost of CPU-time) is less than 20%. Thus, curved perspectives are a good choice in virtual reality applica- tions where ultra-wide-angle lenses have to be used. Key words: Curved perspectives – Wide- angle lenses – Orthogonal projection – Stereo- graphic projection – Cylindrical projection – Mollweide projection – Eckert projection When we take a picture of a scene by means of a photographic camera, we have a projection of 3D space onto a flat image plane. The focus of the cam- era only has influence on the boundary (size) of the image: the smaller the focus, the greater the amount of the scene which is visible. According to the rules of classic perspective, the images of straight lines ap- pear as straight lines. Such perspectives usually pro- duce a realistic output for a focus f ≥ 30 mm. They become unrealistic and misleading for 18 mm < f < 30 mm and differ considerably from the images our brain normally produces with even smaller foci. Dis- tances and angles can then hardly be estimated (see Figs. 1d, 2, and 14b), and surfaces of revolution like spheres (Figs. 11b and 13a) or cylinders (Fig. 9b) ap- pear unnatural, for example. The human eye can only see sharply within a narrow cone with an apex angle ϕ (“field of vision angle” or “fovy-angle”) of about one degree (see Fig. 6a, Baier 1959). When we look at a scene that requires a large fovy-angle (i.e., a small focus 1 ), our eye balls will not stay fixed. Rather, they rotate quickly in order to produce several images of details of the scene (“centers of interest”). Our brain then unites those partial images into one “impression”. This im- pression only corresponds to images produced by a classic perspective with fovy-angles less than 30 ◦ (see, for example, Strubecker 1969). Objects outside the cone of revolution, with the eye point C as apex, the principal projection ray as axis, and the apex an- gle of approximately 30 ◦ , appear unrealistic. 2 Figure 2 illustrates how animated sequences can lead to unwanted “movements” of points in the scene (e.g., the front corners of the table). The viewer stands close to a corner of a normal-sized room and rotates the horizontal main projection ray through 8 ◦ each time. Note the relative path curves of the front corners of the table (hyperbolas) and the unnatural appearance of the corresponding 90 ◦ angles. Then compare this with Fig. 8. 1 The focus f corresponds indirectly proportionally to the fovy- angle ϕ (Hech 1974) ϕ 3.4 ◦ 6.7 ◦ 13.4 ◦ 26.5 ◦ 38 ◦ f 400 mm 200 mm 100 mm 50 mm 35 mm ϕ 46 ◦ 53 ◦ 63 ◦ 78 ◦ f 28 mm 24 mm 20 mm 16 mm 2 The section line of this cone with the projection plane π is a circle with diameter ≈ d, where d is the distance Cπ of the perspective. In the image, object-silhouettes outside this circle are distorted unnaturally. The Visual Computer (1999) 15:365–376 c  Springer-Verlag 1999