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. Where do they go? • Line at Infinity • Plane + line at infinity = Projective Plane

Points at Infinity

• Artists’ use vanishing points. • Pappus’ Theorem -

Consider parallel lines A1B2, A2B1.

Does the theorem hold? A3 A2 A1

B2 B1 B3

History of Math R. L. Herman Fall 2021 12/18 • Desargues - line at infinity. • Look at a plane • Add parallel lines. Where do they go? • Line at Infinity • Plane + line at infinity = Projective Plane

Points at Infinity

• Artists’ use vanishing points. • Pappus’ Theorem -

Consider parallel lines A1B2, A2B1.

Does the theorem hold? A3 A2 A1

B2 B1 B3

History of Math R. L. Herman Fall 2021 12/18 • Desargues - line at infinity. • Look at a plane • Add parallel lines. Where do they go? • Line at Infinity • Plane + line at infinity = Projective Plane

Points at Infinity

• Artists’ use vanishing points. • Pappus’ Theorem -

Consider parallel lines A1B2, A2B1.

Does the theorem hold? A3 A2 A1

B2 B1 B3

History of Math R. L. Herman Fall 2021 12/18 • Look at a plane • Add parallel lines. Where do they go? • Line at Infinity • Plane + line at infinity = Projective Plane

Points at Infinity

• Artists’ use vanishing points. • Pappus’ Theorem -

Consider parallel lines A1B2, A2B1.

Does the theorem hold? A3 A2 A1 • Desargues - line at infinity.

B2 B1 B3

History of Math R. L. Herman Fall 2021 12/18 • Add parallel lines. Where do they go? • Line at Infinity • Plane + line at infinity = Projective Plane

Points at Infinity

• Artists’ use vanishing points. A3 A2 • Pappus’ Theorem - A1

Consider parallel lines A1B2, A2B1. Does the theorem hold? • Desargues - line at infinity. B2 B1 • Look at a plane B3

History of Math R. L. Herman Fall 2021 12/18 • Add parallel lines. Where do they go? • Line at Infinity • Plane + line at infinity = Projective Plane

Points at Infinity

• Artists’ use vanishing points. A3 A2 • Pappus’ Theorem - A1

Consider parallel lines A1B2, A2B1. Does the theorem hold? • Desargues - line at infinity. B2 B1 • Look at a plane B3

History of Math R. L. Herman Fall 2021 12/18 Points at Infinity

A3 A2 • Artists’ use vanishing points. A1 • Pappus’ Theorem -

Consider parallel lines A1B2, A2B1. Does the theorem hold? B 2 B 1 B • Desargues - line at infinity. 3 • Look at a plane • Add parallel lines. Where do they go? • Line at Infinity • Plane + line at infinity = Projective Plane Line at infinity

History of Math R. L. Herman Fall 2021 12/18 • Add point at infinity, 1 real projective line, RP . • Topologically a circle! Intersection: y = b, y = mx : • We can map the circle to . b R x = m .

Projective Line

• Consider the real line, R.

∞ − + ∞ 0

History of Math R. L. Herman Fall 2021 13/18 • Topologically a circle! Intersection: y = b, y = mx : • We can map the circle to . b R x = m .

Projective Line

• Consider the real line, R. • Add point at infinity, 1 real projective line, RP .

∞ − + ∞ 0

History of Math R. L. Herman Fall 2021 13/18 • We can map the circle to R.

Intersection: y = b, y = mx : b x = m .

Projective Line

• Consider the real line, R. • Add point at infinity, 1 real projective line, RP . • Topologically a circle!

∞ − + ∞ 0 ∞

− +

0 History of Math R. L. Herman Fall 2021 13/18 Intersection: y = b, y = mx : b x = m .

Projective Line

• Consider the real line, R. y • Add point at infinity, 1 real projective line, RP . • Topologically a circle! • We can map the circle to R.

∞ − + ∞ x 0 ∞

− +

0 History of Math R. L. Herman Fall 2021 13/18 Projective Line

• Consider the real line, R. y • Add point at infinity, 1 real projective line, RP . • Topologically a circle! • We can map the circle to R.

∞ − + ∞ x 0 ∞

− + Intersection: y = b, y = mx : b x = m . 0 History of Math R. L. Herman Fall 2021 13/18 Projective Line

• Consider the real line, R. y • Add point at infinity, 1 real projective line, RP . • Topologically a circle! • We can map the circle to R.

∞ − + ∞ x 0 ∞

− + Intersection: y = b, y = mx : b x = m . 0 History of Math R. L. Herman Fall 2021 13/18 Homogeneous Coordinates

• Point on line: (x, y, z) • All points on line map to (X , Y ) in the plane. z • (X , Y ) are called homogeneous z = 1 (X , Y ) Y coordinates. (x, y, z) X • Points on line are multiples, (x 0, y 0, z0) = λ(x, y, z). y • Point on plane: ( x , y , 1), or z z x x y X = , Y = . z z

History of Math R. L. Herman Fall 2021 14/18 • Slicing with planes, like Alberti’s veil, one gets projections of the curve.

Curves

• Curve in plane, Y = X 2. • Translates to y x 2 = . z z • This is a surface in (x, y, z)-space,

x 2 = yz.

Figure 3: Surface x 2 − yz.

History of Math R. L. Herman Fall 2021 15/18 Curves

• Curve in plane, Y = X 2. • Translates to y x 2 = . z z • This is a surface in (x, y, z)-space,

x 2 = yz.

• Slicing with planes, like Alberti’s veil, one gets projections of the curve. Figure 3: Surface x 2 − yz.

History of Math R. L. Herman Fall 2021 15/18 Curves

• Curve in plane, Y = X 2. • Translates to y x 2 = . z z • This is a surface in (x, y, z)-space,

x 2 = yz.

• Slicing with planes, like Alberti’s veil, one gets projections of the curve. Figure 3: Surface x 2 − yz.

History of Math R. L. Herman Fall 2021 15/18 Projective Sphere

• Map points on a plane to points z on surface of unit sphere, S2. z = 1 • Lines through South Pole uniquely intersect the plane and sphere. • All points mapped except y (0, 0, 0). This point can be x mapped to the line at infinity. • Lines through origin are points of the real projective plane, 2 RP . Figure 4: Stereographic Projection

History of Math R. L. Herman Fall 2021 16/18 Looking into the Veil

Veil z

(0, −4, 4)

0 02 (x, 0, z) y = x

y

(x0, y 0, 0) x

4x0 4y 0 x = , z = y 0 + 4 y 0 + 4

4x 4z x0 = , y 0 = 4 − z 4 − z

Figure 5: Problems 8.4.2-8.4.4 History of Math R. L. Herman Fall 2021 17/18 Looking into the Veil

Veil z

(z − 2)2 (0, −4, 4) x2 + = 1 4 0 02 (x, 0, z) y = x

y

(x0, y 0, 0) x

4x0 4y 0 x = , z = y 0 + 4 y 0 + 4

4x 4z x0 = , y 0 = 4 − z 4 − z

Figure 5: Problems 8.4.2-8.4.4 History of Math R. L. Herman Fall 2021 17/18 Perspective Drawing

Looking at conics from a different perspective: The parabola z = −x 2 looks like an ellipse.

Vanishing Point z = −x 2 y

x

z

In the 1600’s mathematicians had other mathematics to attend to. So, we return to geometry in the 1800’s.

History of Math R. L. Herman Fall 2021 18/18