3 GEOMETRY AND GRAPHICS – EXAMPLES AND EXERCISES Preface This textbook is intended for students of subjects ‘Constructive Geometry’ and ‘Computer Graphics’ at the Faculty of Mechanical Engineering at Czech Technical University in Prague. The textbook is organized in the following way: Part I – Geometry consists of many examples on their graphic representation. Because Constructive Geometry makes great demands on the student’s spatial imagination, the three- dimensional models of important geometric problems, which are constructed in this textbook in Monge projection, are available on http://marian.fsik.cvut.cz/~linkeova as ‘Geometry and Graphics – 3D Supplement’. Geometry and Graphics – 3D Supplement is considered to be as an inseparable part of this textbook. Part II – Graphics is devoted to the mathematical modelling of free form curves and surfaces which are used in many CAD/CAM systems nowadays. This part can be useful when modelling free form curves and surfaces in Ferguson, Bézier, Coons, B-spline and NURBS representation. Prague 2013 doc. Ing. Ivana Linkeová, Ph.D.
Transcript
3
GEOMETRY AND GRAPHICS – EXAMPLES AND EXERCISES
Preface
This textbook is intended for students of subjects ‘Constructive Geometry’ and
‘Computer Graphics’ at the Faculty of Mechanical Engineering at Czech Technical University
in Prague. The textbook is organized in the following way:
Part I – Geometry consists of many examples on their graphic representation. Because
Constructive Geometry makes great demands on the student’s spatial imagination, the three-
dimensional models of important geometric problems, which are constructed in this textbook
in Monge projection, are available on http://marian.fsik.cvut.cz/~linkeova as ‘Geometry and
Graphics – 3D Supplement’. Geometry and Graphics – 3D Supplement is considered to be as
an inseparable part of this textbook.
Part II – Graphics is devoted to the mathematical modelling of free form curves and
surfaces which are used in many CAD/CAM systems nowadays. This part can be useful when
modelling free form curves and surfaces in Ferguson, Bézier, Coons, B-spline and NURBS
1.1 Construct the true length of straight line segment AB.
x 12
z2
y
A 2 B 2
A1
B 1
x 12
z2
y
A 2
B 2
A1 B 1
1 1
a) AB || π b) AB || ν
x 12
z2
y
A 2
B 2
A1
B 1
1
x 12
z2
y1
B 1
A1
B 2
A 2
c) AB in general position d) AB in general position
1 MONGE PROJECTION
6
1.2 Construct adjacent views of equilateral triangle ∆ ABC and square � ABCD lying in projecting plane ρ.
x 12
A 2
B 2
a) ∆ ABC ⊂ ρ π
ρ1
n2ρ
x 12
b) ∆ ABC ⊂ ρ ν
p1ρ
ρ2
A1
B 1
x 12
A 2
B 2
c) ABCD ⊂ ρ π
ρ1
n2ρ
x 12
ρ2
A1
B 1
d) ABCD ⊂ ρ ν
p1ρ
1 MONGE PROJECTION
7
1.3 a) Construct adjacent views of a circle k inscribed into square � ABCD lying in projecting plane ρ⊥π.
b) Construct adjacent views of a circle k circumscribed around regular hexagon ABCDEF lying in projecting plane ρ⊥ν.
b)
A1
E1
p 1ρ
ρ 2
b)
B1
p 1ρ
ρ 2
a)
x12
n 2ρ
ρ 1
A2
B2
1 MONGE PROJECTION
8
1.4 Construct adjacent views of a solid, which consists of hemisphere Σ (S, r = SR) and cube ABCDA’B’C’D’. Axis o of the solid, centre S of the hemisphere, front view R2 of the point R of the hemisphere and front view A2 of vertex A of the cube are given.
R2
x 12
S1
S 2
A2
o 2
o 1
2 OBLIQUE PROJECTION
9
2.1 In oblique projection (ω = 135°, q = 3:4), construct oblique view of the cube ABCDA’B’C’D’ with the base ABCD in plane (x, y). The side AB is given. Three circles k, k’ and k” are inscribed into faces BCC’B’, CDD’C’ and A’B’C’D’, respectively. Construct oblique views of circles k, k’ and k”.
y
z
O =A=A =A'1 1
xB=B =B'1 1
2 OBLIQUE PROJECTION
10
2.2 In oblique projection (ω = 120°, q = 2:3), construct oblique view of a cone of revolution with the base in the plane parallel with plane (x, y). Radius of the base (r = 45 mm), centre S of the base and vertex V = O of the cone are given.
x
y
z
O =V=V =S1 1
S
2 OBLIQUE PROJECTION
11
2.3 In oblique projection (ω = 135°, q = 1:2), construct oblique view of a solid. Front view and right side view of the solid are given. Measure the dimension, which you will need.
y3y2x 2
z3
x 3
z2
y
z
x
O
2 OBLIQUE PROJECTION
12
2.4 In military perspective, construct oblique view of a solid. Top view and front view of the solid are given. Measure the dimension, which you will need.
y1
z2
x2
z1x 1
y2
xy
z
O
3 ORTHOGONAL AXONOMETRY, TECHNICAL ISOMETRY
13
3.1 In technical isometry, construct a sphere with the centre S = O, and the radius r = 50 mm. Construct the points of intersection K, L, M of the sphere with x, y and z axis. Construct the curve of intersection e of the sphere and
a) horizontal plane of projection π. b) frontal plane of projection ν.
z
S = O
S = O
z
b)
a)
3 ORTHOGONAL AXONOMETRY, TECHNICAL ISOMETRY
14
3.2 In technical isometry, construct a cube ABCDA’B’C’D’ with the base ABCD in the plane (x, y). Vertex A = O is given. Vertex B lies on positive part of x axis, length of the cube side is 80 mm. Three circles k, k’ and k” are inscribed into faces A’B’C’D’, BCC’B’ and CDD’C’, respectively. Construct technical isometry of circles k, k’ and k”.
z
O =A
3 ORTHOGONAL AXONOMETRY, TECHNICAL ISOMETRY
15
3.3 In technical isometry, construct a detail which is given by technical drawing. Axis of the detail is identical with x axis, centre S of the sphere lies at original O of coordinate system.
z
60
80
O = S
95
50
S
3 ORTHOGONAL AXONOMETRY, TECHNICAL ISOMETRY
16
3.4 In technical isometry, construct a detail which is given by technical drawing. Axis of the detail is identical with y axis, centre S of the sphere lies at original O of coordinate system.
z
O
SPHERE R 40
20
50
160
50
30
S
S
x
y
4 LINEAR PERSPECTIVE
17
4.1 In linear perspective (h, z, H, Dd), construct a squared mesh lying in the ground plane. The rotated top view of the squared mesh is given.
H h
z
A"1
A'1
A1
B"1 C"1 D"1
B'1 C'1 D'1
D 1B 1 C 1
a)
Dd
H
Dd
h
b)
A1
A'1
A"1
B"1
B'1
B1
C"1
C1
z
C'1
4 LINEAR PERSPECTIVE
18
4.2 In linear perspective (h, z, H, Dd), construct a right squared prism ABCDA’B’C’D’. The base ABCD of the prism lies in the ground plane. The prism is placed behind the perspective plane of projection. The rotated top view of the base ABCD and altitude of the prism v = 120 mm are given.
H Dd
h zA
1
D1
B1
C1
4 LINEAR PERSPECTIVE
19
4.3 In linear perspective (h, z, H, Dd), construct a right squared pyramid ABCDV. The base ABCD of the pyramid lies in the ground plane. Pyramid is placed behind the perspective plane of projection. The rotated top view of the base ABCD and altitude of the pyramid v = 120 mm are given.
H Dd
h zA
1
D1
B1
C1
5 KINEMATIC GEOMETRY
20
5.1 The motion is given by trajectories �A, �B of points A, B. a) Construct new positions of given points C, D, E. Draw the trajectory �C, �D, �E. b) Construct new positions of given point C. Construct the tangent lines to the trajectory �
C at all new positions of point C. Draw �C.
τB
B0
A0
C0
D0
E0
τA
a)
τ
A0
B0
A
τBC 0
b)
5 KINEMATIC GEOMETRY
21
5.2 The motion is given by trajectories �A, �B of points A, B. a) Construct new positions of the given straight line segment AB. Construct points of
contact of the straight line segment AB and its envelope (AB) at all new positions. Draw the envelope (AB).
b) Construct fixed centrode p and moving centrode h at the given instant.
τ
A0
B0
A
τB
a)
τ
A0
B0
A
τB
b)
5 KINEMATIC GEOMETRY
22
5.3 The motion is given by trajectory �A of point A and point envelope (b) of straight line b. Construct new positions of the given circle k with centre C. Construct the tangent lines to the trajectory �C at all new positions of point C. Draw the trajectory �C. Construct the points of contact of the circle k and its envelope (k) at all new positions of circle k. Draw the envelope (k). Construct fixed centrode p.
τ
A0
A
C0 (b
)
SτA
k0
5 KINEMATIC GEOMETRY
23
5.4 The motion is given by envelopes (a), (b) of straight lines a, b. Construct new positions of the given point C. Construct the tangent lines to the trajectory �C at all new positions of the given point C. Draw the trajectory �C of point C. Construct fixed centrode p and moving centrode h at the given instant.
(a)
(b)
C 0
P0
5 KINEMATIC GEOMETRY
24
5.5 Cycloidal motion is given by centrodes p, h. a) Construct new positions of the given point A. Construct the tangent lines to the
trajectory �A at all new positions of the given point A. Draw the trajectory �A of the point A.
b) Construct new positions of the given straight line segment a. Construct the points of contact of the straight line segment a and its envelope (a) at all new positions of straight line a. Draw the envelope (a).
A0h0
a0
h0
p p
a) b)
5 KINEMATIC GEOMETRY
25
5.6 Involute motion is given by centrodes p, h. Construct new positions of the given point C. Construct the tangent lines to the trajectory �C at all new positions of given point C. Draw the trajectory �C of point C. Construct new positions of the given straight line a. Construct the points of contact of the straight line a and its envelope (a) at all new positions of straight line a. Draw the envelope (a).
C0
h0a0
p
5 KINEMATIC GEOMETRY
26
5.7 Hypocycloidal motion is given by centrodes p, h (rp : rh = 2:1) a) Construct new positions of the given point C. Construct the tangent lines at all new
positions of point C. Draw the trajectory �C. b) Construct new positions of the given straight line AB. Construct the points of contact
of the straight line AB and its envelope (AB) at all new positions of straight line AB. Draw the envelope (AB).
A0
B0
p
C 0
h0
a)
A0
B0
p
h0
b)
5 KINEMATIC GEOMETRY
27
5.8 Epicycloidal motion is given by centrodes p, h (rp : rh = 3:1). Construct new positions of the given points A, B, C. Draw the trajectories �A, �B, �C.
C0
A0
B0
h 0
p
R45
R15
6 SURFACES OF REVOLUTION
28
6.1 A surface of revolution κ is given by its generating curve k and axis of revolution o⊥�. In Monge projection, construct the missing view of the point A lying on the surface κ. At the point A construct the tangent plane τ of the surface κ and the normal line n to the surface κ.
x12k 2 k 1
o2
S2
S 1o 1
A1
b)
x12k 2
A2
o2 o 1
k 1
a)
6 SURFACES OF REVOLUTION
29
6.2 The surface of revolution κ is given by its generating curve k and axis of revolution o⊥�. In the given half-plane �, construct the principal half-meridian m of the surface κ. Use Monge projection.
x 12
k2
o 2
σ1
o1
k1
6 SURFACES OF REVOLUTION
30
6.3 Two surfaces of revolution are given by their half-meridian m, m’ and axis of revolution o⊥�, o’⊥�. Construct the intersection of these surfaces. Indicate the visibility. Use Monge projection.
b) x 12
o 2 o 1
o' 2 o' 1
m2
m1
m' 2 m
' 1
x 12
o 2
o 1
a)
o' 2
o'1
m2
m1
m' 1
m' 2
6 SURFACES OF REVOLUTION
31
6.4 A conical surface of revolution (principal half-meridian m, axis o⊥�) and a torus (principal half-meridian m’, axis o’⊥�) are given. In Monge projection, construct the intersection of the torus and the conical surface of revolution. Indicate the visibility.
x 12
o 2
m'2
m1o1
o'2
m2
m'1o'1
6 SURFACES OF REVOLUTION
32
6.5 A conical surfaces of revolution κ and a cylindrical surface of revolution κ’ are given. In Monge projection, construct the intersection of the surfaces κ and κ’. Indicate the visibility.
o 2
o'2
κ2
o'1o1
x 12
κ'2
κ'1
κ1
6 SURFACES OF REVOLUTION
33
6.6 A conical surface of revolution κ’ and a cylindrical surface of revolution κ are given. In Monge projection, construct the intersection of the surfaces κ’ and κ. Indicate the visibility.
x 12
o' 2
o 2κ 2
κ' 2
κ 1κ' 1
o 1o' 1
b)
x 12a)
o' 2
o 2κ 2
κ' 2
κ 1
κ' 1o 1
o' 1
7 HELIX
34
7.1 A cylinder of revolution κ (bases k, k', axis o is identical with z-axis) and the point A lying on the base k are given. In military perspective, construct one and half thread of right-handed helix h generated by screw motion of the given point A. The axis of the screw motion is identical with the axis o of cylinder of revolution κ. The lead is to be equal to 120 mm.
xy
z = o
o 1
k
k'
κ
A = A1
7 HELIX
35
7.2 In Monge projection, construct one thread of right-handed helix h generated by screw motion of the given point A. Construct point of intersection B of the helix h and plane σ and point of intersection C of the helix h and plane �. Determine how many solutions the problem has. Axis of the screw motion o⊥π, the lead of screw motion v = 120 mm, plane σ⊥π and plane �⊥ν are given.
o 1
o2
A 2
A1
x 12
σ1
ρ 2
7 HELIX
36
7.3 A cylinder of revolution κ (bases k, k', axis o is identical with z-axis) and the point A lying on the base k are given. In military perspective, construct one thread of left-handed helix h (axis o⊥π, left-handed, generating point A, parameter of screw motion v0) and one thread of the surface generated by tangent lines of the helix h. Construct only the part of surface between the helix and horizontal plane of projection �.
z = o
o1
k
k'
κ
A = A1
v0
x
y
7 HELIX
37
7.4 In Monge projection, construct point of intersection B of helix h (generating point A, axis o⊥π, parameter v0) and plane σ. Construct tangent line of helix h at the given point A and at the point of intersection B both.
a) Orientation of the screw motion is left-handed. b) Orientation of the screw motion is right-handed.
b) x12
o2
o 1
v 0
A2
A1
σ2
x12
A2
o2
o 1
a)
A1
v 0
σ 1
8 HELICOIDAL SURFACES
38
8.1 In Monge projection, construct the missing view of a point A lying on helicoidal surface κ (generating curve k, axis o⊥π, parameter of screw motion v0) and tangent plane τ of helicoidal surface κ at the point A.
a) Orientation of the screw motion is left-handed. b) Orientation of the screw motion is right-handed. Choose one solution only.
o 1o2
x12
v 0
k 2 k1
S2
S 1
b)A
2
x12
o2 v 0 o 1
A1
a)
k1
k 2
8 HELICOIDAL SURFACES
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8.2 In Monge projection, construct the missing view of point A lying on generating curve k
(circle with centre S) of helicoidal surface κ (generating curve k, axis o⊥π, parameter of right-handed screw motion v0). Construct normal section k’ of helicoidal surface κ by plane σ⊥ν. Construct point A’ lying on normal section k’ corresponding to the given point A. Construct the tangent plane τ of helicoidal surface κ at point A’.
x 12
o1
v0
S1
k 1
S2
k 2
S1
k 1
σ 2
A1
o 2
8 HELICOIDAL SURFACES
40
8.3 In Monge projection, construct the normal section k’ of helicoidal surface κ (generating
curve k, axis o⊥π, parameter of left-handed screw motion v0).
x 12
o1
v0
B = S1
B2
A2
A1
S 2
1
k 1
k 2
o2 C2
C 1
σ1
8 HELICOIDAL SURFACES
41
8.4 In Monge projection, construct the principle half-meridian of helicoidal surface
κ (generating curve k, axis o⊥π, parameter of right-handed screw motion v0) in given half-plane σ.
x 12
o2
σ1
o1
v0
S1
k 1
S2 k 2
8 HELICOIDAL SURFACES
42
8.5 In Monge projection, construct the normal section k’ of helicoidal surface κ (generating curve k, axis o⊥π, parameter of left-handed screw motion v0).
x 12
o1
v0
B = S1
B2
A2
A1
S 2
1
k 1
k 2
o2
C2
C 1
σ 2
8 HELICOIDAL SURFACES
43
8.6 In Monge projection, construct the principal meridian k’, k” of helicoidal surface κ (generating curve k, axis o⊥π, parameter of right-handed screw motion v0) in given plane σ.
x 12
v0
o2
P2
Q1
k2
k1
σ1o1
P1
Q2
8 HELICOIDAL SURFACES
44
8.7 In Monge projection, construct the normal section k’ of helicoidal surface κ (generating curve k, axis o⊥π, parameter of right-handed screw motion v0).
o2 σ2
v0
o1
x 12
P2
Q2
k2
P1
Q1
9 ENVELOPE SURFACES
45
9.1 In Monge projection, construct the characteristic curve k of envelope surface κ which is generated by rotation of plane σ around axis o⊥π. Construct top view and front view of envelope surface κ.
a) Rotated plane σ⊥π (given by its top view σ1 and frontal trace n2σ) is parallel to axis o.
b) Rotated plane σ (given by triangle ∆ABC) intersects axis o.
A1
B1
C1
C2
A2
B2
o2
x12
o 1
b)
o2
x12
o 1
a)
σ 1
n 2σ
9 ENVELOPE SURFACES
46
9.2 In Monge projection, construct the characteristic curve k of envelope surface κ generated by translation of sphere Σ along path p. Construct top view and front view of envelope surface κ between horizontal plane of projection π and frontal plane of projection ν.
x 12
p 2
p 1
S2
S1
Σ2
Σ1
9 ENVELOPE SURFACES
47
9.3 In Monge projection, construct the characteristic curve k of envelope surface κ generated by rotating sphere Σ around the axis o⊥π. Construct principle half-meridian m in given plane σ. Construct top view and front view of envelope surface κ.
o2
o1 σ1
x 12
S2
S1
Σ2
Σ1
9 ENVELOPE SURFACES
48
9.4 In Monge projection, construct the characteristic curve k of envelope surface κ generated by a screw motion of the plane σ. Screw motion is given by axis o⊥π and parameter v0.
a) Plane σ⊥π (given by its top view σ1 and frontal trace n2σ) is parallel to axis o.
Orientation of the screw motion is left-handed. b) Plane σ (given by its from view σ2 and frontal trace p1
σ) intersects axis o. Orientation of the screw motion is right-handed.
o2
x12
o 1
b)
v 0
p 1σ
σ 2
o2
x12
o 1
a) σ 1
n 2σ
v 0
9 ENVELOPE SURFACES
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9.5 In Monge projection, construct the characteristic curve k of envelope surface κ generated by screwing sphere Σ. Screw motion is given by axis o⊥π and parameter v0 of right handed screw motion. Generating sphere Σ is placed in basic position (S2 ∈ o2).
x 12
S2
S1
Σ2
Σ1
o2
o1
v0
σ1
9 ENVELOPE SURFACES
50
9.6 In Monge projection, construct the characteristic curve k of envelope surface κ generated by rotation of surface of revolution Σ around axis o⊥π. Construct principal half-meridian m in the given plane σ.
Σ2
Σ1
S1 S'1
x 12
o2
o1 σ1
S2 S'2
9 ENVELOPE SURFACES
51
9.7 In Monge projection, construct the characteristic curve k of envelope surface κ generated by rotation of cylindrical surface of revolution Σ around axis o⊥π. Construct the principal half-meridian m in the given plane σ.
x 12
S'1
S'2
S1
S2
Σ2
Σ1
o2
o1 σ1
10 DEVELOPABLE SURFACES
52
10.1 The right circular cylinder κ (directrix k, axis o⊥π) and section plane σ⊥ν are given. Develop the surface of cylinder κ and curve of intersection e = κ ∩ σ.
x12
o
σ 2
k'2
p 11
κ 2
κ 1
e 2
o 2
k 2
σ
k =k
' =e
11
1
10 DEVELOPABLE SURFACES
53
10.2 The intersection of two right circular cylinders Σ (directrix k, axis o⊥π) and Σ’ (directrix k’, axis o’ || ν) degenerates. Develop the surfaces of cylinders Σ and Σ’.
A0B0
C0
D0
A2
x12
o 2
o' 2
o =S 1
o' 1
k 2
k' 2
k'' 2
B2
C2
D2
S 2
S'2
S'' 2
S'1
S'' 1
1
Σ 2
Σ' 2
10 DEVELOPABLE SURFACES
54
10.3 A right circular cone κ (directrix k, vertex V) and section plane σ⊥ν are given. Develop the surface of cylinder κ and curve of intersection e = κ ∩ σ.
x12
V
σ 2 p 11
κ 2
κ 1
e 2
V 2
k 2
σ
k 1
10 DEVELOPABLE SURFACES
55
10.4 An oblique circular cylinder κ (diretrix k, centre line ST) and point A are given.. Construct top and front views of the cylinder κ. Construct the normal section e of the cylinder κ with the plane which passes through point A. Develop the surface of cylinder κ and normal section e.
x12
S1
T 1
A =
A1
S2
2T
2
k 1
B2
B2
B0
10 DEVELOPABLE SURFACES
56
10.5 An oblique circular cone κ (diretrix k, vertex V) is given. Construct top view and front view of the cone κ. Develop the surface of the cone κ.
x12
S1
V 1
S2
V2
A2
A1
k 1k2
A0
V0
11 TRANSITION DEVELOPABLE SURFACES
57
11.1 A polyline ABC and circle k are given. In Monge projection, construct the smooth developable transition surface between polyline ABC and circle k. Develop the transition surface.
x 12
A2 B 2
S2
k2
S1
k1
A1 B1
A0 B0
11 TRANSITION DEVELOPABLE SURFACES
58
11.2 A polyline ABCD and circle k are given. In Monge projection, construct the smooth developable transition surface between polyline ABCD and circle k. Develop the transition surface. In military perspective, construct oblique view of transition the surface.
x 12
z2
y1
A 1
A =B2 2
B 1C1
1D
C =D2 2
S1
S =E2k 2
E1
2
k1
y
z
x
O
A0
B0
11 TRANSITION DEVELOPABLE SURFACES
59
11.3 A rectangle ABCD and circle k are given. In Monge projection, construct the smooth developable transition surface between rectangle ABCD and circle k. Develop the transition surface.
A0
B0
x 12
z2
y1
S1
S 2 k 2
k1
B 1C1
1
E1
A 1 D
A =B2 2 C =D2 2
E2
11 TRANSITION DEVELOPABLE SURFACES
60
11.4 A hexagon m and circle k are given. In Monge projection, construct the smooth developable transition surface between hexagon m and circle k. Develop the transition surface. In military perspective, construct oblique view of the transition surface.
