Computer Graphics Through OpenGL: From Theory to

Experiments, Second Edition

Appendix A

Figure A.1: Perceiving objects with a point camera and a plane film.

Figure A.2: Perceiving points, lines and planes by projection.

Figure A.3: (a) Projective points are radial lines (b) A projective line consists of allprojective points on a radial plane: projective points P and P’ belong to the projectiveline L, while P’’ does not. Keep the distinction in mind that, though we have labeled theplane L, the projective line L actually consists of all the projective points, e.g., P andP’, that lie on this plane, and is different from the plane itself.

Figure A.4: (a) Radial lines corresponding to projective points P and P’ are containedin a unique radial plane corresponding to the projective line L (b) Radial planescorresponding to projective lines L and L’ intersect in a unique radial line correspondingto the projective point P.

Figure A.5: The coordinates of any point on P, except the origin, can be used as itshomogeneous coordinates – four possibilities are shown.

Figure A.6: Real point p on the plane z = 1 is associated with the projective pointφ(p). Projective point Q, lying on the plane z = 0, is not associated with any real point.

Figure A.7: The real points p and p’ travel along parallel lines l and l’. Associatedprojective points φ(p) and φ(p’) travel with p and p’.

Figure A.8: φ(p) travels along L and φ(p’) along L’. L and L’ meet at P’’.

Figure A.9: The line l (= projective point P) is parallel to lines in l. P is said to bethe point at infinity along the equivalence class l of parallel lines.

Figure A.10: Power lines y = 2; z = 2 projected onto the planes (a) z = 1 and (b)x = 1. Red lines depict light rays. The x-axis corresponds to the projective point P.

Figure A.11: Screenshotof turnFilm1.cpp.

Figure A.12: Transform these snapshots on the plane z = 1 to the plane x = 1. Somepoints on the plane z = 1 are shown with their xy coordinates. Labels correspond toitems of Exercise A.7.

Figure A.13: Answer toExercise A.7(h).

Figure A.14: Point p ofradial line l lies on radialplane q, implying that llies on q; point p’ of l’doesn't lie on q, implyingthat no point of l’, otherthan the origin, lies on q.

Figure A.15: Lifting a parabola drawn on the real plane z = 1 to the projective plane.

Figure A.16: Thecoordinate patch Bcontaining P in P2 is inone-to-one correspondencewith the rectangle Wcontaining p in R2 (a fewpoints in W and theircorresponding projectivepoints are shown).

Figure A.17: IdentifyingP1 with a circle.

Figure A.18: Projective transformation of a car (purely conceptual!).

Figure A.19: (a) A segment s on R2 and its lifting S (b) fM transforms s to s and Sto hM(S), while s’ is the intersection of hM(S) with z = 1.

Figure A.20: Rectangle r is transformed to the trapezoid hM(r).

Figure A.21: (a) Projective transformation hM maps rectangle r to trapezoidr’ = hM(r) (b) r’ is the “same” as r’’, the picture of r captured on a film along x = 1.

Figure A.22: Aligningplane p with p’ by aparallel displacement, sothat their respectivedistances from the originare equal, followed by arotation.

Figure A.23: A snapshottransformation to aparallel plane is equivalentto a scaling by a constantfactor in all directions.

Figure A.24: Venndiagram of transformationclasses of R2.

Figure A.25: Transforming the trapezoid q on z = 1 to the rectangle (bold) q’.

Figure A.26: The squareq is mapped to thequadrilateral q’ by asnapshot transformation.