Projective Geometry Fall 2021 - R. L. Herman Perspective Drawing • Art - Perspective Drawing • Before Renaissance- no illusion of depth and space. • 13th century Italian masters used shadowing. • Mathematics of perspective Lengths not preserved. Angles not preserved. Figure 1: c.1308-1311 History of Math R. L. Herman Fall 2021 1/18 Filippo Brunelleschi (1377-1446) • Architect • First to describe linear perspective. • Experimented (1415-1420) using a panel with a grid of squares and a plaque with a hole at eye level. • Drawings of the Baptistry in Florence, Place San Giovanni and other Florence landmarks. • Later his method was studied by Alberti and Da Vinci. History of Math R. L. Herman Fall 2021 2/18 Leon Battista Alberti (1404-1472) • Alberti's Veil Transparent cloth on a frame, Good for actual scenes not imaginary ones. • Basic principles: 1. A straight line in perspective remains straight. 2. Parallel lines either remain parallel or converge to a point. • Drawing a square-tiled floor, solved by Alberti (1436). History of Math R. L. Herman Fall 2021 3/18 • Choose one tile. • Extend diagonal. • Intersections determine the horizontals. Alberti's Method • Align nonhorizontal lines equally along base, converging to one point on the horizon. Vanishing Point Horizon Base History of Math R. L. Herman Fall 2021 4/18 • Extend diagonal. • Intersections determine the horizontals. Alberti's Method • Align nonhorizontal lines equally along base, converging to one point on the horizon. • Choose one tile. Vanishing Point Horizon Base History of Math R. L. Herman Fall 2021 4/18 • Intersections determine the horizontals. Alberti's Method • Align nonhorizontal lines equally along base, converging to one point on the horizon. • Choose one tile. • Extend diagonal. Vanishing Point Horizon Base History of Math R. L. Herman Fall 2021 4/18 Alberti's Method • Align nonhorizontal lines equally along base, converging to one point on the horizon. • Choose one tile. • Extend diagonal. • Intersections determine the horizontals. Vanishing Point Horizon Base History of Math R. L. Herman Fall 2021 4/18 Alberti's Method • Align nonhorizontal lines equally along base, converging to one point on the horizon. • Choose one tile. • Extend diagonal. • Intersections determine the horizontals. Vanishing Point Horizon Base History of Math R. L. Herman Fall 2021 4/18 Alberti's Method • Align nonhorizontal lines equally along base, converging to one point on the horizon. • Choose one tile. • Extend diagonal. • Intersections determine the horizontals. Vanishing Point Horizon Base History of Math R. L. Herman Fall 2021 4/18 Alberti's Method • Align nonhorizontal lines equally along base, converging to one point on the horizon. • Choose one tile. • Extend diagonal. • Intersections determine the horizontals. Vanishing Point Horizon Base History of Math R. L. Herman Fall 2021 4/18 Desargues' Projective Geometry1 A A2 3 • Mathematics behind Alberti's Veil: A1 Family of lines (light rays) through C1 C3 a point (eye) plus a plane (veil). C2 B1 • Recall Pappus' Theorem: A1; A2; B2 B3 A3; collinear; B1; B2; B3; collinear; A2 A3 then, so are C1; C2; C3: A1 • Blaise Pascal (1623-1662) at 16 generalized to conics. B1 B3 • Desargues (1640) Projective B2 Geometry only relies on a straight Figure 2: Pappus' and Pascal's edge. Theorems. 1 Two centuries ahead of his time. History of Math R. L. Herman Fall 2021 5/18 • b) Extend Aa, Bb, Cc center of perspectivity P. • a) Extend pairs, AC-ac, AB-ab, etc. • Points are collinear. • What if two sides are parallel? • Need Projective plane. Girard Desargues (1591-1661) • Architect in Paris, Lyon and engineer. • Desargues' Theorem in appendix of book on perspective, by friend A Abraham Bosse (1602{1676). Two triangles are a) in perspective a b axially if and only if they are b) in B perspective centrally. c C History of Math R. L. Herman Fall 2021 6/18 • a) Extend pairs, AC-ac, AB-ab, etc. • Points are collinear. • What if two sides are parallel? • Need Projective plane. Girard Desargues (1591-1661) • Architect in Paris, Lyon and engineer. • Desargues' Theorem in appendix of book on perspective, by friend A Abraham Bosse (1602{1676). Two triangles are a) in perspective a b axially if and only if they are b) in B perspective centrally. • b) Extend Aa, Bb, Cc c center of perspectivity P. C History of Math R. L. Herman Fall 2021 6/18 • a) Extend pairs, AC-ac, AB-ab, etc. • Points are collinear. • What if two sides are parallel? • Need Projective plane. Girard Desargues (1591-1661) • Architect in Paris, Lyon and engineer. • Desargues' Theorem in appendix of book on perspective, by friend A Abraham Bosse (1602{1676). Two triangles are a) in perspective a b axially if and only if they are b) in B perspective centrally. P • b) Extend Aa, Bb, Cc c center of perspectivity P. C History of Math R. L. Herman Fall 2021 6/18 • Points are collinear. • What if two sides are parallel? • Need Projective plane. Girard Desargues (1591-1661) • Architect in Paris, Lyon and engineer. A • Desargues' Theorem in appendix of a book on perspective, by friend b B Abraham Bosse (1602{1676). P Two triangles are a) in perspective axially if and only if they are b) in c perspective centrally. • b) Extend Aa, Bb, Cc C center of perspectivity P. • a) Extend pairs, AC-ac, AB-ab, etc. History of Math R. L. Herman Fall 2021 6/18 • Points are collinear. • What if two sides are parallel? • Need Projective plane. Girard Desargues (1591-1661) • Architect in Paris, Lyon and engineer. A • Desargues' Theorem in appendix of a book on perspective, by friend b Abraham Bosse (1602{1676). B Two triangles are a) in perspective axially if and only if they are b) in c perspective centrally. • b) Extend Aa, Bb, Cc C center of perspectivity P. • a) Extend pairs, AC-ac, AB-ab, etc. History of Math R. L. Herman Fall 2021 6/18 • Points are collinear. • What if two sides are parallel? • Need Projective plane. Girard Desargues (1591-1661) • Architect in Paris, Lyon and A engineer. a • Desargues' Theorem in appendix of b B book on perspective, by friend Abraham Bosse (1602{1676). Two triangles are a) in perspective c axially if and only if they are b) in perspective centrally. C • b) Extend Aa, Bb, Cc center of perspectivity P. • a) Extend pairs, AC-ac, AB-ab, etc. History of Math R. L. Herman Fall 2021 6/18 • Points are collinear. • What if two sides are parallel? • Need Projective plane. Girard Desargues (1591-1661) • Architect in Paris, Lyon and A engineer. a • Desargues' Theorem in appendix of b B book on perspective, by friend Abraham Bosse (1602{1676). Two triangles are a) in perspective c axially if and only if they are b) in perspective centrally. C • b) Extend Aa, Bb, Cc center of perspectivity P. • a) Extend pairs, AC-ac, AB-ab, etc. History of Math R. L. Herman Fall 2021 6/18 • What if two sides are parallel? • Need Projective plane. Girard Desargues (1591-1661) • Architect in Paris, Lyon and A engineer. a • Desargues' Theorem in appendix of b B book on perspective, by friend Abraham Bosse (1602{1676). Two triangles are a) in perspective c axially if and only if they are b) in perspective centrally. C • b) Extend Aa, Bb, Cc center of perspectivity P. • a) Extend pairs, AC-ac, AB-ab, etc. • Points are collinear. Axis of Perspectivity History of Math R. L. Herman Fall 2021 6/18 • Need Projective plane. Girard Desargues (1591-1661) • Architect in Paris, Lyon and A engineer. a • Desargues' Theorem in appendix of b B book on perspective, by friend Abraham Bosse (1602{1676). Two triangles are a) in perspective c axially if and only if they are b) in perspective centrally. C • b) Extend Aa, Bb, Cc center of perspectivity P. • a) Extend pairs, AC-ac, AB-ab, etc. • Points are collinear. • What if two sides are parallel? Axis of Perspectivity History of Math R. L. Herman Fall 2021 6/18 Girard Desargues (1591-1661) • Architect in Paris, Lyon and A engineer. a • Desargues' Theorem in appendix of b B book on perspective, by friend Abraham Bosse (1602{1676). Two triangles are a) in perspective c axially if and only if they are b) in perspective centrally. C • b) Extend Aa, Bb, Cc center of perspectivity P. • a) Extend pairs, AC-ac, AB-ab, etc. • Points are collinear. • What if two sides are parallel? Axis of Perspectivity • Need Projective plane. History of Math R. L. Herman Fall 2021 6/18 Invariance of the Cross Ratio Lengths and angles are not preserved. A A0 B B0 C C 0 D D0 For any four points on a line, AC : AD is invariant, or BC BD AC AD A0C 0 A0D0 : = : : BC BD B0C 0 B0D0 History of Math R. L. Herman Fall 2021 7/18 Projective Geometry Rebirth in 1800's. Perspective 1. Parallel lines meet at a pt. 2. Lines map to lines. 3. Conics map to conics. Example: Train tracks. vanishing point horizon History of Math R. L. Herman Fall 2021 8/18 One Point Perspective - Find the Vanishing Points History of Math R. L. Herman Fall 2021 9/18 Two Point Perspective - Find the Vanishing Points History of Math R. L. Herman Fall 2021 10/18 Two Point Perspective - Find the Vanishing Points History of Math R. L. Herman Fall 2021 10/18 Two Point Perspective Vanishing Points History of Math R. L. Herman Fall 2021 11/18 Two Point Perspective Vanishing Points History of Math R. L. Herman Fall 2021 11/18 • Desargues - line at infinity. • Look at a plane • Add parallel lines.