Synthetic projective methods used for solving problems in...

Synthetic projective methods used for solvingproblems in geometry

Michal Zamboj

Mathematical Institute of the Charles University, MUUK

WDS–MPrague

17.6.2017

Mgr. Michal Zamboj Synthetic projective methods

K. G. Ch. von Staudt

Karl Georg Christian vonStaudt1798-1867German mathematicianstudent of Gauss1835 professor at Erlangen

Selected works:1847 Geometrie der Lage

1856 - 60 Beiträge zur Geometrie der Lage

“. . . Yet man is no mole. Infinite feelers radiate from the windowsof his soul, whose wings touch the fixed stars. The angel of lightin him created for the guidance of eye-life an independentsystem, a radiant geometry, a visual space, codified in 1847 bya new Euclid, by the Erlangen professor, Georg von Staudt, inhis immortal Geometrie der Lage published in the quaint andancient Nürnberg of Albrecht Dürer. . . . ”

The preface of Synthetic Projective Geometry, by G. B. Halsted

Von Staudt’s contribution in projective geometry:projective geometry (geometry of position) derivedsynthetically - "Wurf"synthetic construction of the algebraic structure of fieldthe fundamental theorem of projective geometry (in R)synthetic construction of imaginary elements

Method of projection

Central perspectivity preserves WurfAny perspectivity or a composition of two or moreperspectivities is a projectivity

ExampleWurfs of collinear points ABCD and BADC are equal.

[ A′B′C′D

A′B′C′D(A)[ OXC′C

OXC′C(B′)[ BADC

ABCD Z BADC

ABCD(O)

[ A′B′C′D

OXC′C(B′)[ BADC

ABCD Z BADC

ABCD(O)

[ A′B′C′D

OXC′C(B′)[ BADC

ABCD Z BADC

ABCD(O)

[ A′B′C′D

OXC′C(B′)[ BADC

ABCD Z BADC

ABCD(O)

[ A′B′C′D

OXC′C(B′)[ BADC

ABCD Z BADC

Projective space

Definitions of the (real) projective space:projective extension of the real spaceaxiomatically - built on the incidence propertyexistentially - as a real vector space

affine plane→ projective plane

afiine points [x , y ]→ projective points (x1, x2, x0); x0 6= 0affine directions (infinite points) (x , y)→ projective points (x1, x2,0)

projective plane→ affine plane

Substracting one line l∞ from P2

projective affine

von Staudt’s constructions

A choice of the projective coordinate system of anone-dimensional form - elements 0,1,∞.

The sum of elements A,B (Wurfs 01∞A; 01∞B)

A choice of the projective coordinate system of anone-dimensional form - elements 0,1,∞.

The product of elements A,B (Wurfs 01∞A; 01∞B)

Cross-ratio

Definition (Cross-ratio in projective coordinates)

Let a(a1,a0),b(b1,b0), c(c1, c0), d(d1,d0) be four elements ofthe same one-dimensional form. Then

(a,b; c,d) =

∣∣∣∣a1 a0c1 c0

∣∣∣∣ ∣∣∣∣b1 b0d1 d0

∣∣∣∣∣∣∣∣b1 b0c1 c0

∣∣∣∣ ∣∣∣∣a1 a0d1 d0

∣∣∣∣ =(a1c0 − a0c1)(b1d0 − b0d1)

(b1c0 − b0c1)(a1d0 − a0d1)

is said to be the cross ratio of elements a,b, c,d (in this order).

Cross-ratio in homogenous coordinates

Let A(a1,a0),B(b1,b0), C(c1, c0), D(d1,d0) be four points inRP1. Then

(A,B;C,D) =

∣∣∣∣a1 a0c1 c0

∣∣∣∣ ∣∣∣∣b1 b0d1 d0

∣∣∣∣∣∣∣∣b1 b0c1 c0

∣∣∣∣ ∣∣∣∣a1 a0d1 d0

∣∣∣∣Let A(a1,a2,a0),B(b1,b2,b0), C(c1, c2, c0), D(d1,d2,d0) be fourpoints on a line l in RP2 and let O(o1,o2,o0) be a point not on l .Then

(A,B;C,D) =[OAC] · [OBD]

[OBC] · [OAD]

D is the harmonic conjugate of C with respect to A and B.(A,B;C,D) = −1

One elementary theorem

Let ABHG be a trapezoid, where AB‖HG and E be theintersection point of AG and BF . The intersection point I ofdiagonals AH and BG lies on the median of the triangle ABEthrough E .

One elementary theorem

Let ABHG be a trapezoid, where AB‖HG and E be theintersection point of AG and BF . The intersection point I ofdiagonals AH and BG lies on the median of the triangle ABEthrough E .

A centroid of a triangle (with respect to a line)

With a little help from GeoGebra

Menelaus’s theorem

Theorem (Menelaus’s theorem)

If in a triangle ABC with cutting points K ∈ BC,L ∈ CA,M ∈ AB,the points K ,L,M are collinear, then we have

(A,B;M) · (B,C;K ) · (C,A;L) = 1

(A, B; M) · (B, C; K ) · (C, A; L) ?= 1

with respect to a line

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= 1

ABMX ′(W ′)[ DBKY ′, CALZ ′

(W ′)[ CDKY ′

(D, B; K , Y ′) · (B, C; K , Y ′) · (C, D; K , Y ′) ?= 1

−→DK ·

−→BY

−→BK ·−→DY·−→BK ·−→CY

−→CK ·

−→BY·−→CK ·

−→DY

−→DK ·

−→CY

(A, B; M) · (B, C; K ) · (C, A; L) ?= 1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= 1

(W ′)[ CDKY ′

(D, B; K , Y ′) · (B, C; K , Y ′) · (C, D; K , Y ′) ?= 1

−→DK ·

−→BY

−→BK ·−→DY·−→BK ·−→CY

−→CK ·

−→BY·−→CK ·

−→DY

−→DK ·

−→CY

(A, B; M) · (B, C; K ) · (C, A; L) ?= 1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= 1

(W ′)[ CDKY ′

(D, B; K , Y ′) · (B, C; K , Y ′) · (C, D; K , Y ′) ?= 1

−→DK ·

−→BY

−→BK ·−→DY·−→BK ·−→CY

−→CK ·

−→BY·−→CK ·

−→DY

−→DK ·

−→CY

(A, B; M) · (B, C; K ) · (C, A; L) ?= 1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= 1

(W ′)[ CDKY ′

(D, B; K , Y ′) · (B, C; K , Y ′) · (C, D; K , Y ′) ?= 1

−→DK ·

−→BY

−→BK ·−→DY·−→BK ·−→CY

−→CK ·

−→BY·−→CK ·

−→DY

−→DK ·

−→CY

(A, B; M) · (B, C; K ) · (C, A; L) ?= 1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= 1

(W ′)[ CDKY ′

(D, B; K , Y ′) · (B, C; K , Y ′) · (C, D; K , Y ′) ?= 1

−→DK ·

−→BY

−→BK ·−→DY·−→BK ·−→CY

−→CK ·

−→BY·−→CK ·

−→DY

−→DK ·

−→CY

(A, B; M) · (B, C; K ) · (C, A; L) ?= 1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= 1

(W ′)[ CDKY ′

(D, B; K , Y ′) · (B, C; K , Y ′) · (C, D; K , Y ′) ?= 1

−→DK ·

−→BY

−→BK ·−→DY·−→BK ·−→CY

−→CK ·

−→BY·−→CK ·

−→DY

−→DK ·

−→CY

Ceva’s theorem

Theorem (Ceva’s theorem)

If in a triangle ABC with cutting points K ∈ BC,L ∈ CA,M ∈ AB,the lines AK ,BL,CM are concurrent, then we have

(A,B;M) · (B,C;K ) · (C,A;L) = −1

Ceva’s theorem

(A, B; M) · (B, C; K ) · (C, A; L) ?= −1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= −1

ABMX ′(S)

[ ALCX , BCKY ′(S)

[ LCAY

(A, L; C, X) · (L, C; A.Y ) · (C, A; L, Z ′) ?= −1

[SAC]·[SLX ][SLC]·[SAX ]

· [SLA]·[SCY ][SCA]·[SLY ]

· [SCL]·[SAZ′ ][SAL]·[SCZ′ ] ·

([SLZ′ ][SLZ′ ]

= − [SLX ]·[SAZ′ ][SAX ]·[SLZ′ ] ·

[SCY ]·[SLZ′ ][SLY ]·[SCZ′ ]

= − (L,A;X,Z′)(L,C;Y ,Z′)

LAXZ ′(X′)[ LBSW ′; LCYZ ′

(Y ′)[ LBSW ′

= − (L,B;S,W ′)(L,B;S,W ′) = −1

Ceva’s theorem

(A, B; M) · (B, C; K ) · (C, A; L) ?= −1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= −1

ABMX ′(S)

[ LCAY

(A, L; C, X) · (L, C; A.Y ) · (C, A; L, Z ′) ?= −1

([SLZ′ ][SLZ′ ]

= − [SLX ]·[SAZ′ ][SAX ]·[SLZ′ ] ·

= − (L,A;X,Z′)(L,C;Y ,Z′)

(Y ′)[ LBSW ′

= − (L,B;S,W ′)(L,B;S,W ′) = −1

Ceva’s theorem

(A, B; M) · (B, C; K ) · (C, A; L) ?= −1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= −1

ABMX ′(S)

[ LCAY

(A, L; C, X) · (L, C; A.Y ) · (C, A; L, Z ′) ?= −1

([SLZ′ ][SLZ′ ]

= − [SLX ]·[SAZ′ ][SAX ]·[SLZ′ ] ·

= − (L,A;X,Z′)(L,C;Y ,Z′)

(Y ′)[ LBSW ′

= − (L,B;S,W ′)(L,B;S,W ′) = −1

Ceva’s theorem

(A, B; M) · (B, C; K ) · (C, A; L) ?= −1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= −1

ABMX ′(S)

[ LCAY

(A, L; C, X) · (L, C; A.Y ) · (C, A; L, Z ′) ?= −1

([SLZ′ ][SLZ′ ]

= − [SLX ]·[SAZ′ ][SAX ]·[SLZ′ ] ·

= − (L,A;X,Z′)(L,C;Y ,Z′)

(Y ′)[ LBSW ′

= − (L,B;S,W ′)(L,B;S,W ′) = −1

Ceva’s theorem

(A, B; M) · (B, C; K ) · (C, A; L) ?= −1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= −1

ABMX ′(S)

[ LCAY

(A, L; C, X) · (L, C; A.Y ) · (C, A; L, Z ′) ?= −1

· [SCL]·[SAZ′ ][SAL]·[SCZ′ ]

[SLZ′ ][SLZ′ ]

= − [SLX ]·[SAZ′ ][SAX ]·[SLZ′ ] ·

= − (L,A;X,Z′)(L,C;Y ,Z′)

(Y ′)[ LBSW ′

= − (L,B;S,W ′)(L,B;S,W ′) = −1

Ceva’s theorem

(A, B; M) · (B, C; K ) · (C, A; L) ?= −1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= −1

ABMX ′(S)

[ LCAY

(A, L; C, X) · (L, C; A.Y ) · (C, A; L, Z ′) ?= −1

([SLZ′ ][SLZ′ ]

= − [SLX ]·[SAZ′ ][SAX ]·[SLZ′ ] ·

= − (L,A;X,Z′)(L,C;Y ,Z′)

(Y ′)[ LBSW ′

= − (L,B;S,W ′)(L,B;S,W ′) = −1

Ceva’s theorem

(A, B; M) · (B, C; K ) · (C, A; L) ?= −1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= −1

ABMX ′(S)

[ LCAY

(A, L; C, X) · (L, C; A.Y ) · (C, A; L, Z ′) ?= −1

([SLZ′ ][SLZ′ ]

= − [SLX ]·[SAZ′ ][SAX ]·[SLZ′ ] ·

= − (L,A;X,Z′)(L,C;Y ,Z′)

(Y ′)[ LBSW ′

= − (L,B;S,W ′)(L,B;S,W ′) = −1

Ceva’s theorem

(A, B; M) · (B, C; K ) · (C, A; L) ?= −1

(A, B; M, X ′) · (B, C; K .Y ′) · (C, A; L, Z ′) ?= −1

ABMX ′(S)

[ LCAY

(A, L; C, X) · (L, C; A.Y ) · (C, A; L, Z ′) ?= −1

([SLZ′ ][SLZ′ ]

= − [SLX ]·[SAZ′ ][SAX ]·[SLZ′ ] ·

= − (L,A;X,Z′)(L,C;Y ,Z′)

(Y ′)[ LBSW ′

= − (L,B;S,W ′)(L,B;S,W ′) = −1

Pappus’s theorem

Theorem (Pappus’s theorem)

Let A,B,C be three points on a straight line and let A′,B′,C′ bethree points on another line. If the lines AB′,CA′,BC′ intersectthe lines BA′,AC′,CB′, respectively, then the three points ofintersection are collinear.

Pappus’s theorem

One special easy-to-prove case

Pappus’s theorem

X = AB′ ∩ BA′; Z = BC′ ∩ CB′

Y = AC′ ∩ CA′?∈ XZ

Let AC′ ∩ XZ = Y1, CA′ ∩ XZ = Y2

ABPQ(X)

[ B′A′PQ′

ABPQ(C′)

[ Y1ZQ′Q

B′A′PQ′(C′)

[ ZY2QQ′

but Y1ZQ′Q Z ZY1QQ′

therefore ZY1QQ′ Z ZY2QQ′

Y1 = Y2 = Y ∈ XZ

Pappus’s theorem

X = AB′ ∩ BA′; Z = BC′ ∩ CB′

Y = AC′ ∩ CA′?∈ XZ

ABPQ(X)

[ B′A′PQ′

ABPQ(C′)

[ Y1ZQ′Q

B′A′PQ′(C′)

[ ZY2QQ′

Y1 = Y2 = Y ∈ XZ

Pappus’s theorem

X = AB′ ∩ BA′; Z = BC′ ∩ CB′

Y = AC′ ∩ CA′?∈ XZ

ABPQ(X)

[ B′A′PQ′

ABPQ(C′)

[ Y1ZQ′Q

B′A′PQ′(C′)

[ ZY2QQ′

Y1 = Y2 = Y ∈ XZ

Pappus’s theorem

X = AB′ ∩ BA′; Z = BC′ ∩ CB′

Y = AC′ ∩ CA′?∈ XZ

ABPQ(X)

[ B′A′PQ′

ABPQ(C′)

[ Y1ZQ′Q

B′A′PQ′(C′)

[ ZY2QQ′

Y1 = Y2 = Y ∈ XZ

Pappus’s theorem

X = AB′ ∩ BA′; Z = BC′ ∩ CB′

Y = AC′ ∩ CA′?∈ XZ

ABPQ(X)

[ B′A′PQ′

ABPQ(C′)

[ Y1ZQ′Q

B′A′PQ′(C′)

[ ZY2QQ′

Y1 = Y2 = Y ∈ XZ

Pappus’s theorem

X = AB′ ∩ BA′; Z = BC′ ∩ CB′

Y = AC′ ∩ CA′?∈ XZ

ABPQ(X)

[ B′A′PQ′

ABPQ(C′)

[ Y1ZQ′Q

B′A′PQ′(C′)

[ ZY2QQ′

Y1 = Y2 = Y ∈ XZ

Pappus’s theorem

X = AB′ ∩ BA′; Z = BC′ ∩ CB′

Y = AC′ ∩ CA′?∈ XZ

ABPQ(X)

[ B′A′PQ′

ABPQ(C′)

[ Y1ZQ′Q

B′A′PQ′(C′)

[ ZY2QQ′

Y1 = Y2 = Y ∈ XZ

Pappus’s theorem

X = AB′ ∩ BA′; Z = BC′ ∩ CB′

Y = AC′ ∩ CA′?∈ XZ

ABPQ(X)

[ B′A′PQ′

ABPQ(C′)

[ Y1ZQ′Q

B′A′PQ′(C′)

[ ZY2QQ′

Y1 = Y2 = Y ∈ XZ

References

Paraphrasing Cayley: ". . . projective geometry is all geometry."

1847 VON STAUDT, K. G. CH.: Geometrie der Lage1856-60 VON STAUDT, K. G. CH.: Beiträge zur Geometrie der Lage

1964 COXETER, H. S. M.: Projective Geometry1944-45 HLAVATÝ, V.: Projektivní geometrie I.-II.

2011 RICHTER - GEBERT, J.: Perspectives on ProjectiveGeometry

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