SlideShare ist ein Scribd-Unternehmen logo
1 von 50
Downloaden Sie, um offline zu lesen
Drawing chords in perspective
H.S.M. Coxeter, Projective Geometry 2nd ed.
“6. Still working in the Euclidean plane, draw a line segment OC, take G two
thirds of the way along it, and E two-fifths of the way from G to C.

  (For instance, make the distance in centimeters OG = 10, GE = 2, EC = 3.)

If the segment OC represents a stretched string, tuned to the note C, the same
string stopped at E or G will play the other notes of the major triad. By drawing
a suitable triangle, verify experimentally that H(OE,CG).

           (Such phenomena explain our use of the word harmonic.)”
Cremona, Elements of Projective Geometry, Oxford University Press, 1913

           Matthews, Projective Geometry, Longmans, Green and co., 1914

               Veblen and Young, Projective Geometry Volume 2, 1918

     Young, Projective Geometry, Carus Mathematical Monographs, MAA, 1930

         Baer, Linear Algebra and Projective Geometry, Academic Press, 1952

     Seidenberg, Lectures in Projective Geometry, D Van Nostrand and Co, 1962

            Pedoe, Introduction to Projective Geometry, Macmillan, 1963

     Fishback, Projective and Euclidean Geometry 2ed, John Wiley & Sons, 1969

          Bennett, Affine and Projective Geometry, John Wiley & Sons, 1995

  Kadison and Kromann, Projective Geometry and Modern Algebra, Birkhauser, 1996

Beutelspacher and Rosenbaum, Projective Geometry: From Foundations to Applications,
                         Cambridge University Press, 1998

      Casse, Projective Geometry, an Introduction, Oxford University Press, 2006
Linnaeus Wayland Dowling, Projective Geometry,
      McGraw-Hill Book Co, Inc., 1917 (Forgotten Books)

Notes.- The idea of four harmonic points, or harmonic division, was known to the
early Greek geometers, but who first invented it is not definitely known. Apollonius
of Perga (247 BC) mentions it is his book on conic sections.

The harmonic property of a complete quadrangle is contained in the Collections of
Pappus (300AD). It was made the foundation for Von Staudt’s Geometric der Lage,
1847.

Three cords consisting of the same substance and having the same size and tension,
and whose lengths are in harmonic progression, will vibrate in harmony when struck
in unison. The name harmonic is probably due to that fact.
Harmonic sets in mathematics




                                                            Drawing in perspective


Lewis Goupy, Brook Taylor. 1720




                                  Harmonics in music
Harmonic sets defined by ratios

                   Ceva’s Theorem




                   Menelaeus’ Theorem
Harmonic sets defined by ratios
Harmonic sets defined by ratios




      Harmonic Set: H(AC,BD)
Harmonic sets defined by ratios




      Harmonic Set: H(AC,BD)
Drawing chords in perspective
Harmonic set from a circle and its tangent




            Harmonic Set: H(AC,BD)
Harmonic set from two circles and their tangents




              Harmonic Set: H(AC,BD)
Harmonic set on Euler’s line




                              A = circumcenter
                              B = centroid
                              C = 9-point circle center
                              D = orthocenter




     Harmonic Set: H(AC,BD)
Harmonic sets in projective geometry…
              What is projective geometry?
                      A non-Euclidean geometry which developed
                      out of the mathematics of perspective drawing.


                      A study of geometric properties that are
                      invariant under projections.




Abraham Bosse, 1665
                  A real projective plane is an   Leon Battista Alberti, 1435
   extension of the real Euclidean plane, extended
          by strategically adding points and a line at infinity.
Euclidean Geometry                          Projective Geometry
Constructions with compass and             Constructions with just a straightedge
straightedge
Parallel Postulate: Given a line and a      There are no parallel lines: Any two
point not on the line, there is just one    lines are incident with a unique point.
line through the point parallel to the line
A study of properties invariant under      A study of properties invariant under
rigid motions, like length, angle, area    projections, like…
Euclidean to Projective
Harmonic sets defined by quadrilaterals
Drawing chords in perspective
Definition: A projectivity is a composition of perspectivities.
Theorem : A projectivity preserves harmonic sets.
Drawing chords in perspective
Drawing chords in perspective
Drawing chords in perspective
Drawing chords in perspective
Drawing chords in perspective
Drawing chords in perspective
Definition: A projective collineation
is a composition of perspective
collineations.
Collineations in perspective drawing
Consider translation from the red square to the blue square in
perspective.
Collineations in perspective drawing
Translation in perspective is a perspective collineation.
Collineations in perspective drawing
Reflection in perspective is a perspective collineation.




                      English Boy Using Reflection in Mirror in Foyer of Grand Hotel
                      to Fix His Tie Photographic Print by Alfred Eisenstaedt
Collineations in perspective drawing
180˚ rotation in perspective is a perspective collineation.
Collineations in perspective drawing
45˚ rotation in perspective is NOT a perspective collineation.
Collineations in perspective drawing
45˚ rotation in perspective is a projective collineation, the
composition of two reflections.
Harmonic sets and perspective drawing




then H(BD, AC) ?
Harmonic sets and perspective drawing




then H(BD, AC) ?
Harmonic sets and perspective drawing
Harmonic sets and perspective drawing
Harmonic sets and perspective drawing
Harmonic sets and the harmonic sequence
Drawing chords in perspective
Drawing chords in perspective
Drawing chords in perspective
Harmonic sets, the harmonic sequence
and the harmonic mean of two numbers




   1/3       1/2     1      1/3       1/2      1
   1/4       1/3     1/2    2/4       2/3      2/2
   1/5       1/4     1/3    3/5       3/4      3/3
   1/6       1/5     1/4    4/6       4/5      4/4
 1/(n+2)   1/(n+1)   1/n   n/(n+2)   n/(n+1)   n/n
Harmonic sets and its relationship with
           harmonics in music
Definition: The frequency of a sound wave is the number of
cycles per second, measured in Hertz.
Definition: The pitch of a sound is the perception of frequency.
Harmonic sets and its relationship with…
           harmonics in music
Definition: A harmonic of a sound wave is an integer multiple
of the fundamental frequency of the sound wave.
Harmonic sets and its relationship with…
           harmonics in music
Definition: Overtones are frequencies higher than the
fundamental frequency.

Many musical instruments are
created to have harmonic
overtones.

The human voice can create
overtones.
Harmonic sets, the harmonic sequence, the harmonic
   mean of two numbers, and harmonics in music




       1       55/55      1/2     55/110     1/3      55/165
       1      110/110     2/3     110/165    2/4      110/220
       1      165/165     3/4     165/220    3/5      165/275
       1      220/220     4/5     220/275    4/6      220/330
       1      275/275     5/6     275/330    5/7      275/385
      n/(n)     …       n/(n+1)     …       n/(n+2)     …
Drawing chords in perspective
Harmonic sets and its relationship with
        harmonics in music
                             Just intervals

1   1/2   1/3     1:1/3 = 3:1            1:1/2 = 2:1     1:2:3
                   2 octaves               1 octave
1   2/3   2/4     1:1/2 = 2:1            1:2/3 = 3:2     2:3:4
                    1 octave             Perfect fifth
1   3/4   3/5     1:3/5 = 5:3            1:3/4 = 4:3     3:4:5
                  Major sixth           Perfect fourth
1   4/5   4/6     1:2/3 = 3:2            1:4/5 = 5:4     4:5:6
                  Perfect fifth          Major third
1   5/6   5/7     1:5/7 = 7:5            1:5/6 = 6:5     5:6:7
                 Subminor fifth          Minor third
1   7/8   6/8     1:3:4 = 4:3           1:7/8 = 8:7      6:7:8
                 Perfect fourth      Supermajor second
1   8/9   7/9     1:7/9 = 9:7           1:8/9 = 9:8      7:8:9
                Supermajor third        Major second
Harmonic sets and an android app
      by Stephen Brown

Weitere ähnliche Inhalte

Was ist angesagt?

Trigonometric Ratios
Trigonometric RatiosTrigonometric Ratios
Trigonometric Ratiosliliana1993
 
Math 8 – proofing (direct and indirect)
Math 8 – proofing (direct and indirect)Math 8 – proofing (direct and indirect)
Math 8 – proofing (direct and indirect)Rebekah Andrea Fullido
 
Pairs of Angles Formed by two Parallel Lines Cut by a Transversal-Best lesson...
Pairs of Angles Formed by two Parallel Lines Cut by a Transversal-Best lesson...Pairs of Angles Formed by two Parallel Lines Cut by a Transversal-Best lesson...
Pairs of Angles Formed by two Parallel Lines Cut by a Transversal-Best lesson...Elton John Embodo
 
Rectangular coordinate system
Rectangular coordinate systemRectangular coordinate system
Rectangular coordinate systemCathy Francisco
 
Harmonic division
Harmonic divisionHarmonic division
Harmonic divisionArvee Mae
 
Properties of circle
Properties of circleProperties of circle
Properties of circlerey castro
 
Detailed lesson plan of Similar Triangles in Inductive Method
Detailed lesson plan of Similar Triangles in Inductive MethodDetailed lesson plan of Similar Triangles in Inductive Method
Detailed lesson plan of Similar Triangles in Inductive MethodLorie Jane Letada
 
Law Of Cosines Presentation
Law Of Cosines PresentationLaw Of Cosines Presentation
Law Of Cosines Presentationguest59f920
 
Lesson plan on Linear inequalities in two variables
Lesson plan on Linear inequalities in two variablesLesson plan on Linear inequalities in two variables
Lesson plan on Linear inequalities in two variablesLorie Jane Letada
 
2.7.4 Conditions for Parallelograms
2.7.4 Conditions for Parallelograms2.7.4 Conditions for Parallelograms
2.7.4 Conditions for Parallelogramssmiller5
 
Angles formed by parallel lines cut by transversal
Angles formed by parallel lines cut by transversalAngles formed by parallel lines cut by transversal
Angles formed by parallel lines cut by transversalMay Bundang
 
6.5 Isosceles and Equilateral Triangles
6.5 Isosceles and Equilateral Triangles6.5 Isosceles and Equilateral Triangles
6.5 Isosceles and Equilateral Trianglessmiller5
 
Six Trigonometric Functions Math 9 4th Quarter Week 1.pptx
Six Trigonometric Functions Math 9 4th Quarter Week 1.pptxSix Trigonometric Functions Math 9 4th Quarter Week 1.pptx
Six Trigonometric Functions Math 9 4th Quarter Week 1.pptxMichaelKyleMilan
 
9.4 multiplying and dividing rational expressions
9.4 multiplying and dividing rational expressions9.4 multiplying and dividing rational expressions
9.4 multiplying and dividing rational expressionshisema01
 
Math 8 – mathematics as an axiomatic system
Math 8 – mathematics as an axiomatic systemMath 8 – mathematics as an axiomatic system
Math 8 – mathematics as an axiomatic systemRebekah Andrea Fullido
 

Was ist angesagt? (20)

Trigonometric Ratios
Trigonometric RatiosTrigonometric Ratios
Trigonometric Ratios
 
Math 8 – proofing (direct and indirect)
Math 8 – proofing (direct and indirect)Math 8 – proofing (direct and indirect)
Math 8 – proofing (direct and indirect)
 
Pairs of Angles Formed by two Parallel Lines Cut by a Transversal-Best lesson...
Pairs of Angles Formed by two Parallel Lines Cut by a Transversal-Best lesson...Pairs of Angles Formed by two Parallel Lines Cut by a Transversal-Best lesson...
Pairs of Angles Formed by two Parallel Lines Cut by a Transversal-Best lesson...
 
Rectangular coordinate system
Rectangular coordinate systemRectangular coordinate system
Rectangular coordinate system
 
Harmonic division
Harmonic divisionHarmonic division
Harmonic division
 
Properties of circle
Properties of circleProperties of circle
Properties of circle
 
Conditional Statement.pptx
Conditional Statement.pptxConditional Statement.pptx
Conditional Statement.pptx
 
proving-triangles-are-congruent.ppt
proving-triangles-are-congruent.pptproving-triangles-are-congruent.ppt
proving-triangles-are-congruent.ppt
 
Detailed lesson plan of Similar Triangles in Inductive Method
Detailed lesson plan of Similar Triangles in Inductive MethodDetailed lesson plan of Similar Triangles in Inductive Method
Detailed lesson plan of Similar Triangles in Inductive Method
 
INCIDENCE.pptx
INCIDENCE.pptxINCIDENCE.pptx
INCIDENCE.pptx
 
Law Of Cosines Presentation
Law Of Cosines PresentationLaw Of Cosines Presentation
Law Of Cosines Presentation
 
Lesson plan on Linear inequalities in two variables
Lesson plan on Linear inequalities in two variablesLesson plan on Linear inequalities in two variables
Lesson plan on Linear inequalities in two variables
 
2.7.4 Conditions for Parallelograms
2.7.4 Conditions for Parallelograms2.7.4 Conditions for Parallelograms
2.7.4 Conditions for Parallelograms
 
Angles formed by parallel lines cut by transversal
Angles formed by parallel lines cut by transversalAngles formed by parallel lines cut by transversal
Angles formed by parallel lines cut by transversal
 
6.5 Isosceles and Equilateral Triangles
6.5 Isosceles and Equilateral Triangles6.5 Isosceles and Equilateral Triangles
6.5 Isosceles and Equilateral Triangles
 
Point, Line and plane
Point, Line and planePoint, Line and plane
Point, Line and plane
 
Six Trigonometric Functions Math 9 4th Quarter Week 1.pptx
Six Trigonometric Functions Math 9 4th Quarter Week 1.pptxSix Trigonometric Functions Math 9 4th Quarter Week 1.pptx
Six Trigonometric Functions Math 9 4th Quarter Week 1.pptx
 
Circular permutation
Circular permutationCircular permutation
Circular permutation
 
9.4 multiplying and dividing rational expressions
9.4 multiplying and dividing rational expressions9.4 multiplying and dividing rational expressions
9.4 multiplying and dividing rational expressions
 
Math 8 – mathematics as an axiomatic system
Math 8 – mathematics as an axiomatic systemMath 8 – mathematics as an axiomatic system
Math 8 – mathematics as an axiomatic system
 

Andere mochten auch

Html5 Canvas Drawing and Animation
Html5 Canvas Drawing and AnimationHtml5 Canvas Drawing and Animation
Html5 Canvas Drawing and AnimationMindfire Solutions
 
Perspective Drawing
Perspective DrawingPerspective Drawing
Perspective Drawingdr.f
 
Perspective drawing 1 point a
Perspective drawing 1 point aPerspective drawing 1 point a
Perspective drawing 1 point arangcapan
 
How to Draw Two Point Perspective
How to Draw Two Point PerspectiveHow to Draw Two Point Perspective
How to Draw Two Point PerspectiveBob Black
 
CS193P Lecture 5 View Animation
CS193P Lecture 5 View AnimationCS193P Lecture 5 View Animation
CS193P Lecture 5 View Animationonoaonoa
 
Animate! Figure Drawing for Animation Course
Animate! Figure Drawing for Animation CourseAnimate! Figure Drawing for Animation Course
Animate! Figure Drawing for Animation CourseKath Shackleton
 
City in One Point Perspective Drawing Tutorial
City in One Point Perspective Drawing TutorialCity in One Point Perspective Drawing Tutorial
City in One Point Perspective Drawing TutorialBob Black
 
History Of Perspective
History Of PerspectiveHistory Of Perspective
History Of Perspectivedr.f
 
Value handout
Value handoutValue handout
Value handoutcdpenamtz
 
Line drawing exercise
Line   drawing exerciseLine   drawing exercise
Line drawing exercisecdpenamtz
 
1 point perspective
1 point perspective1 point perspective
1 point perspectiveangiehershey
 
4 watercolor practice 2
4 watercolor practice 24 watercolor practice 2
4 watercolor practice 2cdpenamtz
 
Perspective Presentation
Perspective PresentationPerspective Presentation
Perspective PresentationJeremy Smith
 
11th, Perspective Drawing Workshop, Bim 1, 2011
11th, Perspective Drawing Workshop, Bim 1, 201111th, Perspective Drawing Workshop, Bim 1, 2011
11th, Perspective Drawing Workshop, Bim 1, 2011Artistic Education
 
Braille course
Braille courseBraille course
Braille courseAnpviOnlus
 
1 introduction to watercolor methods
1 introduction to watercolor methods1 introduction to watercolor methods
1 introduction to watercolor methodscdpenamtz
 

Andere mochten auch (20)

Html5 Canvas Drawing and Animation
Html5 Canvas Drawing and AnimationHtml5 Canvas Drawing and Animation
Html5 Canvas Drawing and Animation
 
Perspective Drawing
Perspective DrawingPerspective Drawing
Perspective Drawing
 
Perspective drawing 1 point a
Perspective drawing 1 point aPerspective drawing 1 point a
Perspective drawing 1 point a
 
How to Draw Two Point Perspective
How to Draw Two Point PerspectiveHow to Draw Two Point Perspective
How to Draw Two Point Perspective
 
Burlap wreaths
Burlap wreathsBurlap wreaths
Burlap wreaths
 
CS193P Lecture 5 View Animation
CS193P Lecture 5 View AnimationCS193P Lecture 5 View Animation
CS193P Lecture 5 View Animation
 
Animate! Figure Drawing for Animation Course
Animate! Figure Drawing for Animation CourseAnimate! Figure Drawing for Animation Course
Animate! Figure Drawing for Animation Course
 
City in One Point Perspective Drawing Tutorial
City in One Point Perspective Drawing TutorialCity in One Point Perspective Drawing Tutorial
City in One Point Perspective Drawing Tutorial
 
History Of Perspective
History Of PerspectiveHistory Of Perspective
History Of Perspective
 
Value handout
Value handoutValue handout
Value handout
 
Line drawing exercise
Line   drawing exerciseLine   drawing exercise
Line drawing exercise
 
CM30S - 1.4
CM30S - 1.4CM30S - 1.4
CM30S - 1.4
 
1 point perspective
1 point perspective1 point perspective
1 point perspective
 
4 watercolor practice 2
4 watercolor practice 24 watercolor practice 2
4 watercolor practice 2
 
Perspective drawing
Perspective drawingPerspective drawing
Perspective drawing
 
Perspective Presentation
Perspective PresentationPerspective Presentation
Perspective Presentation
 
11th, Perspective Drawing Workshop, Bim 1, 2011
11th, Perspective Drawing Workshop, Bim 1, 201111th, Perspective Drawing Workshop, Bim 1, 2011
11th, Perspective Drawing Workshop, Bim 1, 2011
 
Braille course
Braille courseBraille course
Braille course
 
1 introduction to watercolor methods
1 introduction to watercolor methods1 introduction to watercolor methods
1 introduction to watercolor methods
 
CM30S - 1.1
CM30S - 1.1CM30S - 1.1
CM30S - 1.1
 

Ähnlich wie Drawing chords in perspective

TRIANGLE CONGRUENCE, SSS, ASA, SAS and AAS
TRIANGLE CONGRUENCE, SSS, ASA, SAS and AASTRIANGLE CONGRUENCE, SSS, ASA, SAS and AAS
TRIANGLE CONGRUENCE, SSS, ASA, SAS and AASMarkChristianBalaald
 
นำเสนอตรีโกณมิติจริง
นำเสนอตรีโกณมิติจริงนำเสนอตรีโกณมิติจริง
นำเสนอตรีโกณมิติจริงkrunittayamath
 
Know it notebook-chapter3
Know it notebook-chapter3Know it notebook-chapter3
Know it notebook-chapter3CarolinaDay3
 
Hcr's formula for regular spherical polygon
Hcr's formula for regular spherical polygonHcr's formula for regular spherical polygon
Hcr's formula for regular spherical polygonHarish Chandra Rajpoot
 
On Polyhedrons
On PolyhedronsOn Polyhedrons
On Polyhedronsiosrjce
 
An Introduction to Trigonometry.pdf
An Introduction to Trigonometry.pdfAn Introduction to Trigonometry.pdf
An Introduction to Trigonometry.pdfJessica Navarro
 
Maths Circle PPT Class10
Maths Circle PPT Class10Maths Circle PPT Class10
Maths Circle PPT Class10Abhey Gupta
 
Conic Section slayerix
Conic Section slayerixConic Section slayerix
Conic Section slayerixAshams kurian
 
207 intro lecture2010
207 intro lecture2010207 intro lecture2010
207 intro lecture2010guest1db8cc6
 
X2 t08 01 circle geometry (2013)
X2 t08 01 circle geometry (2013)X2 t08 01 circle geometry (2013)
X2 t08 01 circle geometry (2013)Nigel Simmons
 

Ähnlich wie Drawing chords in perspective (20)

TRIANGLE CONGRUENCE, SSS, ASA, SAS and AAS
TRIANGLE CONGRUENCE, SSS, ASA, SAS and AASTRIANGLE CONGRUENCE, SSS, ASA, SAS and AAS
TRIANGLE CONGRUENCE, SSS, ASA, SAS and AAS
 
นำเสนอตรีโกณมิติจริง
นำเสนอตรีโกณมิติจริงนำเสนอตรีโกณมิติจริง
นำเสนอตรีโกณมิติจริง
 
Know it notebook-chapter3
Know it notebook-chapter3Know it notebook-chapter3
Know it notebook-chapter3
 
Ellipse.pptx
Ellipse.pptxEllipse.pptx
Ellipse.pptx
 
Hcr's formula for regular spherical polygon
Hcr's formula for regular spherical polygonHcr's formula for regular spherical polygon
Hcr's formula for regular spherical polygon
 
Circles
CirclesCircles
Circles
 
Basics of Stereochemistry
Basics of StereochemistryBasics of Stereochemistry
Basics of Stereochemistry
 
On Polyhedrons
On PolyhedronsOn Polyhedrons
On Polyhedrons
 
ellipse
ellipseellipse
ellipse
 
Paso 4 grupo29
Paso 4 grupo29Paso 4 grupo29
Paso 4 grupo29
 
An Introduction to Trigonometry.pdf
An Introduction to Trigonometry.pdfAn Introduction to Trigonometry.pdf
An Introduction to Trigonometry.pdf
 
Maths Circle PPT Class10
Maths Circle PPT Class10Maths Circle PPT Class10
Maths Circle PPT Class10
 
Conic Section
Conic SectionConic Section
Conic Section
 
Conic Section slayerix
Conic Section slayerixConic Section slayerix
Conic Section slayerix
 
Analytical geometry
Analytical geometryAnalytical geometry
Analytical geometry
 
Modern Geometry Topics
Modern Geometry TopicsModern Geometry Topics
Modern Geometry Topics
 
Circles
CirclesCircles
Circles
 
207 intro lecture2010
207 intro lecture2010207 intro lecture2010
207 intro lecture2010
 
X2 t08 01 circle geometry (2013)
X2 t08 01 circle geometry (2013)X2 t08 01 circle geometry (2013)
X2 t08 01 circle geometry (2013)
 
Case study on circles
Case study on circlesCase study on circles
Case study on circles
 

Mehr von dr.f

Drawing humpty dumpty in perspective
Drawing humpty dumpty in perspectiveDrawing humpty dumpty in perspective
Drawing humpty dumpty in perspectivedr.f
 
Math in wonderland
Math in wonderlandMath in wonderland
Math in wonderlanddr.f
 
Victorian england flatland
Victorian england flatlandVictorian england flatland
Victorian england flatlanddr.f
 
1 d 2d 3d 4d
1 d 2d 3d 4d1 d 2d 3d 4d
1 d 2d 3d 4ddr.f
 
Creating A Culture Of Pedagogical Learning
Creating A Culture Of Pedagogical LearningCreating A Culture Of Pedagogical Learning
Creating A Culture Of Pedagogical Learningdr.f
 
History Of Non Euclidean Geometry
History Of Non Euclidean GeometryHistory Of Non Euclidean Geometry
History Of Non Euclidean Geometrydr.f
 

Mehr von dr.f (6)

Drawing humpty dumpty in perspective
Drawing humpty dumpty in perspectiveDrawing humpty dumpty in perspective
Drawing humpty dumpty in perspective
 
Math in wonderland
Math in wonderlandMath in wonderland
Math in wonderland
 
Victorian england flatland
Victorian england flatlandVictorian england flatland
Victorian england flatland
 
1 d 2d 3d 4d
1 d 2d 3d 4d1 d 2d 3d 4d
1 d 2d 3d 4d
 
Creating A Culture Of Pedagogical Learning
Creating A Culture Of Pedagogical LearningCreating A Culture Of Pedagogical Learning
Creating A Culture Of Pedagogical Learning
 
History Of Non Euclidean Geometry
History Of Non Euclidean GeometryHistory Of Non Euclidean Geometry
History Of Non Euclidean Geometry
 

Kürzlich hochgeladen

Practical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptxPractical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptxKatherine Villaluna
 
Ultra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxUltra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxDr. Asif Anas
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...raviapr7
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxraviapr7
 
How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17Celine George
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17Celine George
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfMohonDas
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRATanmoy Mishra
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxiammrhaywood
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17Celine George
 
How to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 SalesHow to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 SalesCeline George
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfMohonDas
 
How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17Celine George
 
General views of Histopathology and step
General views of Histopathology and stepGeneral views of Histopathology and step
General views of Histopathology and stepobaje godwin sunday
 
5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...CaraSkikne1
 
What is the Future of QuickBooks DeskTop?
What is the Future of QuickBooks DeskTop?What is the Future of QuickBooks DeskTop?
What is the Future of QuickBooks DeskTop?TechSoup
 
Benefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationBenefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationMJDuyan
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17Celine George
 

Kürzlich hochgeladen (20)

Practical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptxPractical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptx
 
Ultra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxUltra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptx
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptx
 
How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdf
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
 
Prelims of Kant get Marx 2.0: a general politics quiz
Prelims of Kant get Marx 2.0: a general politics quizPrelims of Kant get Marx 2.0: a general politics quiz
Prelims of Kant get Marx 2.0: a general politics quiz
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17
 
How to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 SalesHow to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 Sales
 
Personal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdfPersonal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdf
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdf
 
How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17
 
General views of Histopathology and step
General views of Histopathology and stepGeneral views of Histopathology and step
General views of Histopathology and step
 
5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...
 
What is the Future of QuickBooks DeskTop?
What is the Future of QuickBooks DeskTop?What is the Future of QuickBooks DeskTop?
What is the Future of QuickBooks DeskTop?
 
Benefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationBenefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive Education
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17
 

Drawing chords in perspective

  • 2. H.S.M. Coxeter, Projective Geometry 2nd ed. “6. Still working in the Euclidean plane, draw a line segment OC, take G two thirds of the way along it, and E two-fifths of the way from G to C. (For instance, make the distance in centimeters OG = 10, GE = 2, EC = 3.) If the segment OC represents a stretched string, tuned to the note C, the same string stopped at E or G will play the other notes of the major triad. By drawing a suitable triangle, verify experimentally that H(OE,CG). (Such phenomena explain our use of the word harmonic.)”
  • 3. Cremona, Elements of Projective Geometry, Oxford University Press, 1913 Matthews, Projective Geometry, Longmans, Green and co., 1914 Veblen and Young, Projective Geometry Volume 2, 1918 Young, Projective Geometry, Carus Mathematical Monographs, MAA, 1930 Baer, Linear Algebra and Projective Geometry, Academic Press, 1952 Seidenberg, Lectures in Projective Geometry, D Van Nostrand and Co, 1962 Pedoe, Introduction to Projective Geometry, Macmillan, 1963 Fishback, Projective and Euclidean Geometry 2ed, John Wiley & Sons, 1969 Bennett, Affine and Projective Geometry, John Wiley & Sons, 1995 Kadison and Kromann, Projective Geometry and Modern Algebra, Birkhauser, 1996 Beutelspacher and Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998 Casse, Projective Geometry, an Introduction, Oxford University Press, 2006
  • 4. Linnaeus Wayland Dowling, Projective Geometry, McGraw-Hill Book Co, Inc., 1917 (Forgotten Books) Notes.- The idea of four harmonic points, or harmonic division, was known to the early Greek geometers, but who first invented it is not definitely known. Apollonius of Perga (247 BC) mentions it is his book on conic sections. The harmonic property of a complete quadrangle is contained in the Collections of Pappus (300AD). It was made the foundation for Von Staudt’s Geometric der Lage, 1847. Three cords consisting of the same substance and having the same size and tension, and whose lengths are in harmonic progression, will vibrate in harmony when struck in unison. The name harmonic is probably due to that fact.
  • 5. Harmonic sets in mathematics Drawing in perspective Lewis Goupy, Brook Taylor. 1720 Harmonics in music
  • 6. Harmonic sets defined by ratios Ceva’s Theorem Menelaeus’ Theorem
  • 8. Harmonic sets defined by ratios Harmonic Set: H(AC,BD)
  • 9. Harmonic sets defined by ratios Harmonic Set: H(AC,BD)
  • 11. Harmonic set from a circle and its tangent Harmonic Set: H(AC,BD)
  • 12. Harmonic set from two circles and their tangents Harmonic Set: H(AC,BD)
  • 13. Harmonic set on Euler’s line A = circumcenter B = centroid C = 9-point circle center D = orthocenter Harmonic Set: H(AC,BD)
  • 14. Harmonic sets in projective geometry… What is projective geometry? A non-Euclidean geometry which developed out of the mathematics of perspective drawing. A study of geometric properties that are invariant under projections. Abraham Bosse, 1665 A real projective plane is an Leon Battista Alberti, 1435 extension of the real Euclidean plane, extended by strategically adding points and a line at infinity.
  • 15. Euclidean Geometry Projective Geometry Constructions with compass and Constructions with just a straightedge straightedge Parallel Postulate: Given a line and a There are no parallel lines: Any two point not on the line, there is just one lines are incident with a unique point. line through the point parallel to the line A study of properties invariant under A study of properties invariant under rigid motions, like length, angle, area projections, like…
  • 17. Harmonic sets defined by quadrilaterals
  • 19. Definition: A projectivity is a composition of perspectivities.
  • 20. Theorem : A projectivity preserves harmonic sets.
  • 27. Definition: A projective collineation is a composition of perspective collineations.
  • 28. Collineations in perspective drawing Consider translation from the red square to the blue square in perspective.
  • 29. Collineations in perspective drawing Translation in perspective is a perspective collineation.
  • 30. Collineations in perspective drawing Reflection in perspective is a perspective collineation. English Boy Using Reflection in Mirror in Foyer of Grand Hotel to Fix His Tie Photographic Print by Alfred Eisenstaedt
  • 31. Collineations in perspective drawing 180˚ rotation in perspective is a perspective collineation.
  • 32. Collineations in perspective drawing 45˚ rotation in perspective is NOT a perspective collineation.
  • 33. Collineations in perspective drawing 45˚ rotation in perspective is a projective collineation, the composition of two reflections.
  • 34. Harmonic sets and perspective drawing then H(BD, AC) ?
  • 35. Harmonic sets and perspective drawing then H(BD, AC) ?
  • 36. Harmonic sets and perspective drawing
  • 37. Harmonic sets and perspective drawing
  • 38. Harmonic sets and perspective drawing
  • 39. Harmonic sets and the harmonic sequence
  • 43. Harmonic sets, the harmonic sequence and the harmonic mean of two numbers 1/3 1/2 1 1/3 1/2 1 1/4 1/3 1/2 2/4 2/3 2/2 1/5 1/4 1/3 3/5 3/4 3/3 1/6 1/5 1/4 4/6 4/5 4/4 1/(n+2) 1/(n+1) 1/n n/(n+2) n/(n+1) n/n
  • 44. Harmonic sets and its relationship with harmonics in music Definition: The frequency of a sound wave is the number of cycles per second, measured in Hertz. Definition: The pitch of a sound is the perception of frequency.
  • 45. Harmonic sets and its relationship with… harmonics in music Definition: A harmonic of a sound wave is an integer multiple of the fundamental frequency of the sound wave.
  • 46. Harmonic sets and its relationship with… harmonics in music Definition: Overtones are frequencies higher than the fundamental frequency. Many musical instruments are created to have harmonic overtones. The human voice can create overtones.
  • 47. Harmonic sets, the harmonic sequence, the harmonic mean of two numbers, and harmonics in music 1 55/55 1/2 55/110 1/3 55/165 1 110/110 2/3 110/165 2/4 110/220 1 165/165 3/4 165/220 3/5 165/275 1 220/220 4/5 220/275 4/6 220/330 1 275/275 5/6 275/330 5/7 275/385 n/(n) … n/(n+1) … n/(n+2) …
  • 49. Harmonic sets and its relationship with harmonics in music Just intervals 1 1/2 1/3 1:1/3 = 3:1 1:1/2 = 2:1 1:2:3 2 octaves 1 octave 1 2/3 2/4 1:1/2 = 2:1 1:2/3 = 3:2 2:3:4 1 octave Perfect fifth 1 3/4 3/5 1:3/5 = 5:3 1:3/4 = 4:3 3:4:5 Major sixth Perfect fourth 1 4/5 4/6 1:2/3 = 3:2 1:4/5 = 5:4 4:5:6 Perfect fifth Major third 1 5/6 5/7 1:5/7 = 7:5 1:5/6 = 6:5 5:6:7 Subminor fifth Minor third 1 7/8 6/8 1:3:4 = 4:3 1:7/8 = 8:7 6:7:8 Perfect fourth Supermajor second 1 8/9 7/9 1:7/9 = 9:7 1:8/9 = 9:8 7:8:9 Supermajor third Major second
  • 50. Harmonic sets and an android app by Stephen Brown

Hinweis der Redaktion

  1. http://www.music.vt.edu/musicdictionary/appendix/pitch/pitch.html