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An analytical introduction to Descriptive
Geometry
Adrian B. Biran,
Technion – Faculty of Mechanical Engineering
Ruben Lopez-Pulido,
CEHINAV, Polytechnic University of Madrid, Model Basin,
and Spanish Association of Naval Architects
Avraham Banai
Technion – Faculty of Mathematics
Prepared for Elsevier (Butterworth-Heinemann), Oxford, UK
Samples - August 2005
An analytical introduction to descriptive
An analytical
geometry
Pageintroduction
study
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Contents
Preface x
1 Geometric constructions 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Drawing instruments . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 A few geometric constructions . . . . . . . . . . . . . . . . . . 2
1.3.1 Drawing parallels . . . . . . . . . . . . . . . . . . . . . 2
1.3.2 Dividing a segment into two . . . . . . . . . . . . . . . 2
1.3.3 Bisecting an angle . . . . . . . . . . . . . . . . . . . . 2
1.3.4 Raising a perpendicular on a given segment . . . . . . 2
1.3.5 Drawing a triangle given its three sides . . . . . . . . . 2
1.4 The intersection of two lines . . . . . . . . . . . . . . . . . . . 2
1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4.2 Examples from practice . . . . . . . . . . . . . . . . . 2
1.4.3 Situations to avoid . . . . . . . . . . . . . . . . . . . . 2
1.5 Manual drawing and computer-aided drawing . . . . . . . . . 2
i
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ii CONTENTS
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Notations 1
2 Introduction 3
2.1 How we see an object . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Central projection . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.3 Vanishing points . . . . . . . . . . . . . . . . . . . . . 17
2.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Parallel projection . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 A few properties . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 The concept of scale . . . . . . . . . . . . . . . . . . . 25
2.4 Orthographic projection . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 The projection of a right angle . . . . . . . . . . . . . . 28
2.5 The two-sheet method of Monge . . . . . . . . . . . . . . . . . 36
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 The point 51
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CONTENTS iii
3.1 The projections of a point . . . . . . . . . . . . . . . . . . . . 51
3.2 Points in four quadrants . . . . . . . . . . . . . . . . . . . . . 51
3.3 The first angle view . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 The third angle view . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 The straight line 53
4.1 The projections of a straight line . . . . . . . . . . . . . . . . 54
4.2 Special straight lines . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Parallel lines . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 Horizontal lines . . . . . . . . . . . . . . . . . . . . . . 54
4.2.3 Frontal lines . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.4 Profile lines . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.5 Lines perpendicular on π2 . . . . . . . . . . . . . . . . 54
4.2.6 Vertical lines . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.7 Lines perpendicular on π3 . . . . . . . . . . . . . . . . 54
4.3 When a third projection can help . . . . . . . . . . . . . . . . 54
4.4 Intersecting lines . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 The true length of a straight-line segment . . . . . . . . . . . 54
4.6 The traces of a given straight line . . . . . . . . . . . . . . . . 54
4.7 Point on given line . . . . . . . . . . . . . . . . . . . . . . . . 54
4.8 The straight line as a one-dimensional manifold . . . . . . . . 54
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
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An analytical introduction to Descriptive
Geometry
Adrian B. Biran,
Technion – Faculty of Mechanical Engineering
Ruben Lopez-Pulido,
CEHINAV, Polytechnic University of Madrid, Model Basin,
and Spanish Association of Naval Architects
Avraham Banai
Technion – Faculty of Mathematics
Prepared for Elsevier (Butterworth-Heinemann), Oxford, UK
Samples - August 2005
An analytical introduction to descriptive
An analytical
geometry
Pageintroduction
study
1https://online2pdf.com/delete-header-and-footer-in-pdf#
guide.pdfhttps://online2pdf.com/delete-header-and-footer-in-pdf#
to descriptive geometry study guidehttps://online2pdf.com/delete-header-and-footer-in-pdf#
,Page 2 of 159https://online2pdf.com/delete-header-and-footer-in-pdf#
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Contents
Preface x
1 Geometric constructions 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Drawing instruments . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 A few geometric constructions . . . . . . . . . . . . . . . . . . 2
1.3.1 Drawing parallels . . . . . . . . . . . . . . . . . . . . . 2
1.3.2 Dividing a segment into two . . . . . . . . . . . . . . . 2
1.3.3 Bisecting an angle . . . . . . . . . . . . . . . . . . . . 2
1.3.4 Raising a perpendicular on a given segment . . . . . . 2
1.3.5 Drawing a triangle given its three sides . . . . . . . . . 2
1.4 The intersection of two lines . . . . . . . . . . . . . . . . . . . 2
1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4.2 Examples from practice . . . . . . . . . . . . . . . . . 2
1.4.3 Situations to avoid . . . . . . . . . . . . . . . . . . . . 2
1.5 Manual drawing and computer-aided drawing . . . . . . . . . 2
i
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ii CONTENTS
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Notations 1
2 Introduction 3
2.1 How we see an object . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Central projection . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.3 Vanishing points . . . . . . . . . . . . . . . . . . . . . 17
2.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Parallel projection . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 A few properties . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 The concept of scale . . . . . . . . . . . . . . . . . . . 25
2.4 Orthographic projection . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 The projection of a right angle . . . . . . . . . . . . . . 28
2.5 The two-sheet method of Monge . . . . . . . . . . . . . . . . . 36
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 The point 51
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CONTENTS iii
3.1 The projections of a point . . . . . . . . . . . . . . . . . . . . 51
3.2 Points in four quadrants . . . . . . . . . . . . . . . . . . . . . 51
3.3 The first angle view . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 The third angle view . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 The straight line 53
4.1 The projections of a straight line . . . . . . . . . . . . . . . . 54
4.2 Special straight lines . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Parallel lines . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 Horizontal lines . . . . . . . . . . . . . . . . . . . . . . 54
4.2.3 Frontal lines . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.4 Profile lines . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.5 Lines perpendicular on π2 . . . . . . . . . . . . . . . . 54
4.2.6 Vertical lines . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.7 Lines perpendicular on π3 . . . . . . . . . . . . . . . . 54
4.3 When a third projection can help . . . . . . . . . . . . . . . . 54
4.4 Intersecting lines . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 The true length of a straight-line segment . . . . . . . . . . . 54
4.6 The traces of a given straight line . . . . . . . . . . . . . . . . 54
4.7 Point on given line . . . . . . . . . . . . . . . . . . . . . . . . 54
4.8 The straight line as a one-dimensional manifold . . . . . . . . 54
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
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