Descriptive geometry

Surfaces.

Surfaces.
Classification. Determinant. Outline.

Surfaces.
Classification. Determinant. Outline.
Fundamentals:
Surface of geometric solid is multiple of boundary points of a given body.
Boundary point is a center of a sphere which always contains inner and outer points for this body while its radius tend to zero.
To specify a surface in a drawing means to indicate the conditions enabling us to construct each point of this surface.
Representation of a surface:
Analytical method;
Graphical method.
a) framing;
b) kinematic.
Kinematic surface represents a locus of lines moving in space according to a certain law.
These lines, which may be straight or curved, are called generating lines (generatrices).
In general, there is a variety of laws for generating a certain surface. It is desirable to select those laws and shapes of the generating lines which are most simple and convenient both for representing the required surface and solving the problems associated with it.
Determinant of kinematic surface is minimum number of parameters and constraints which univocally represent given surface.
Determinant structure:
Determinant={(geometrical parameters)(algorithm of constructions)}.
Surface on orthographical view can be represented by projections of its determinant.

Surfaces.
Classification.
To systematize properties of surfaces we can classify them by the type of generatrices and character of their motion.

Surfaces.
Classification. Determinant. Outline.
Outline of a Surface.
For a surface to be specified, it is sufficient to have the projections of its directrix (guide line) and adequate information on the method for constructing the generatrix passing through any point of the directrix. But if it is desirable to make the representation more obvious and expressive, then it is advisable to draw also the outline of the surface, several positions of the generatrix, most important lines and points on the surface, etc.
Representation of a surface on a drawing by projections of its determinant can’t provide proper obviousness. It’s reasonable to supplement drawing of a surface by its outline.
Example: a Cone, represented by its determinant:
КВ = {(ℓ‚t) (ℓ∩t, ℓ = R(t)},
Difficult to imagine such a figure without additional explanations.
t2
l2
l1
t1

Surfaces.
Classification. Determinant. Outline.
Outline of a Surface.
П2
S
Contour line
Outline
Projectors
O1
O2
O3
П1
On parallel projection of a surface Ф on the plane П some projectors touch with the surface Ф and form tangent cylindrical projecting surface Q. Tangent curve of Q and Ф surfaces is called Contour Line (Contour Curve), its projection on the plane П – Outline of the surface Ф. This curve can be plane or space.
Outline of a surface is projection of its contour line.
Thus Outline is visibility border for points which belong to a surface.
For instance, the sphere on the planes П1 and П2 has contour curves – circles q2 and m1. They project on the other planes as curves of visibility q1 and m2 – outline curves.
m2
m1
q1
q2

Surfaces.
Classification. Determinant. Outline.
Outline of a surface.
Frontal Visible Contour
Frontal Outline
Horizontal Visible Contour
Horizontal Outline

Surfaces of Revolution. Forming.
Definitions.
Parallel
Meridian
Generatrix
Neck
Axis of revolution
l1
i
m3
m2
m1
o2
o1
o3
i
Meridian plane
A surface of revolution is generated by the revolution of a curved-line or straight-line generatrix about a fixed straight line called the axis of the revolution.
Determinant of the surface of revolution — generatrix and axis of revolution: SR={(l,I)(l=R(I))}
Each point of the generatrix describes a circle. Hence, a plane perpendicular to the axis of revolution cuts this surface in circles. Such circles are called parallels. The biggest parallel is called the equator, the smallest parallel is called the neck of a surface. Neck and equator project to outlines.
A plane passing through the axis of a surface of revolution is termed a meridian plane. The line where a meridian plane intersects a surface of revolution is called the meridian of the surface.
Equator
Cylinder of revolution.
Generatrix – l-straight line ⎜⎜ axis of revolution ί;
n - curve, equally spaced from axis of revolution.
If we imagine the totality of rectilinear generatrices, revolving around the axes, and the totality of generating circles, moving along that axes, then each line of one totality (of one "family" of lines) will intersect all lines of the other totality (of the other "family" of lines). As a result, the mesh of the given surface is obtained. Other surfaces may be thought of in the same way.

Surfaces of Revolution.
General.

Surfaces of Revolution. Forming.
Surface of revolution. Generatrix – Straight Line.
Relative position of Generatrix and Axes of revolution.
CYLINDER
CONE
ONE SHEET HYPERBOLOID OF REVOLUTION
PARALLEL
INTERSECTING
SKEW

Surfaces of Revolution. Forming.
Surface of revolution. Generatrix – Circle.
R>r
R
R=0
TORUS
CLOSED TORUS
SPHERE

Surfaces of Revolution. Examples.
12'
I
I=I'
10''
1'
2'
3'
4'
5'
6'
9''
11''
12''
1''
2''
3''
I=I'
m1
m2
I'
I
1'
12'
2'
11'
10'
9'
8'
3'
7'
6'
5'
4'
5''
4''
3''
2''
1''
6''
10''
9''
8''
7''
12''
11''
For the hyperboloid of revolution the meridian is a hyperbola. If the hyperbola is revolved about its imaginary axis, then we have a hyperboloid of revolution of one sheet. A hyperboloid of revolution of one sheet can also be generated by the revolution of a straight line if the generatrix and the axis of revolution are skew lines.

Surfaces of Revolution. Examples.
C
O
B
I
I'
A
B1
B2
A2
A1
B1
A2
B2
M2
t1
t2
C2
I=I'
A1
C1
M1
Neck

Surfaces of Revolution. Sections by Planes.
O1
42
32
22
12
(1)1
=31
(2)1
=41
11
81
21
31
41
51
61
71
12
22
=82
52
42
=62
32
=72
1'
2'
8'
3'
7'
4'
6'
5'
O1
Minor axes
Major axes
Minor axes

Surfaces of Revolution. Sections by Planes.
S
ELLIPSE – TWO OUTLINES FROM ONE SIDE OF AN APEX ARE INTERSECTED
PARABOLA – PARALLEL TO THE OUTLINE
STRAIGHT LINES – THROUGH APEX ALONG GENERATRIX
HYPERBOLA – TWO OUTLINES FROM DIFFERENT SIDES OF AN APEX ARE INTERSECTED
CIRCLE – PERPENDICULAR TO AN AXES
Cone of revolution. Conical Sections.

Surfaces of Revolution. Sections by Planes.
Point
Ellipse
Circle
Double Parabola
Parabola
Two straight Lines
Hyperbola

Surfaces of Revolution. Sections by Planes.
A
C
B
D
D'
B'
C'
O'
A'
B2
=D2
=O2
C2
C'
A2
C'
D'
O'
O
ELLIPSE

Surfaces of Revolution. Sections by Planes.
O'
M2
=N2
=O2
M'
M'
K2
J2
=L2
L'
J'
N'
K'
L'
Rjl
T2
=P2
T'
σ2
σ'2
T'
P'
HYPERBOLA

Surfaces of Revolution. Sections by Planes.
O'
M2
=N2
=O2
M'
M'
K2
J2
=L2
L'
J'
N'
K'
L'
Rjl
T2
=P2
T'
σ2
σ'2
T'
P'
PARABOLA

Sections of Solids.
Designate all character points of intersection between cutting plane and surfaces of figure.
Locate axes of section 1 – 7 parallel to a cutting line. Mark distances between points in question.
Construct true sizes of full sections from corresponding marked points (for example, А-А section – ellipse from cylinder cutting between points 1 – 7, rectangle from pyramid cutting between points 3 – 6, etc.).
Designate obtained section according to a standard (ISO, GOST, DSTU, etc.).
B
A
A
B-B
2
2'
7
1
2
3
4
5
6
7
7
6
5
7
6
5
4
1
2
2
1
4
3
6'
3'
1'
3'
6'
1'
B
3
4
4'
5
5'
A-A
3
3'
6
6'
1
Combined figure sectioned by a plane

Intersection between a Solid and a Plane.
Three ways how to construct conical section.
1. Using Major and Minor Axis of Ellipse 2. Using Parallels 3. Using Generatrices
A
A
S1
11
31
12
(2)2
32
(4)2
21
41
51
52
S2
B
B
S1
12
(3)2
22
(5)2
31
42
S2
82
102
122
(11)2
21
61
71
121
41
51
81
101
111
91
62
(7)2
(9)2
11
С
С
S1
12
(3)2
22
(5)2
31
42
S2
82
102
122
(11)2
21
121
51
81
101
111
62
(7)2
(9)2
11
61
71
41
91
1
2
5
3
4

Intersection between a Solid and a Plane.
S1
(1)2
32
22
52
31
42
S2
21
51
62
11
61
41
13
33
23
(5)3
43
S3
(6)3
A
A

Surfaces.
Ruled surfaces.
A surface which can be generated by a straight line is called the ruled
surface. Hence, the ruled surface represents a locus of straight lines.
A surface which may be generated only by a curved line will be called the double-
curved surface.

Surfaces.
Developable Ruled surfaces.
Some curved surfaces can be developed so that they coincide completely (with all their points) with a plane without stretching or shrinking.
Such surfaces will be called developable. They comprise only ruled surfaces in which adjacent rectilinear generatrices are parallel or intersect, or are tangent to sphere curve.
All double-curved surfaces and the ruled surfaces which cannot be developed into a plane are called nondevelopable (or warped) surfaces

Surfaces.
Ruled surfaces with one Diretrix.
Conical Surface
m1
m2
A2
A
S
m
M2
M1
S2
l2
l1
A1
S1
A conical surface is generated by a straight line passing through a fixed point and through all the points (in succession) of a curved directing line. This line is also called directrix or guide line. The fixed point S is called the vertex of a conical surface.
S – vertex
m – directrix
ℓ – generatrix
S2
S1

Surfaces.
Ruled surfaces with one Diretrix.
Cylindrical surface
A2
m2
m1
A1
m
l1
A
l
l2
A2
A1
If vertex of conical surface is moved to infinity, then conical surface turns into cylindrical.
A cylindrical surface is generated
by a straight line which is parallel in all its positions to a given straight line
and passes in succession through all points of a curved directing line.
S – vertex → ∞
m – directrix
ℓ - generatrix

Surfaces.
Ruled surfaces with one Diretrix.
n, k – border lines
l1
m1
B1
m2
l2
B2
A1
A2
l
A
m
The surface called a surface with a cuspidal edge or torse surface is generated by a rectilinear generatrix performing continuous motion and touching a space curve at all its positions.
This space curve serves as the directrix for the surface under consideration and is called the cuspidal edge.
Obviously, by specifying the projections of the cuspidal edge, we can specify a surface in the drawing.
Determinant of a torse surface:
Curve line m as a directrix.
Straight line ℓ as a generatrix.
n
k
B

Surfaces.
Ruled surfaces with one Diretrix.
Beveled Gear surface Spur Gear surface

Surfaces.
Ruled surfaces with two Diretrices.
where,
l— generatrix,
m, n —directrices;
σ — plane director.
Ruled surface can be determined by two directrices (guide lines) and plane director (plane of parallelism).
Determinant of a surface :

Surfaces.
Ruled surfaces with two Diretrices.
CYLINDROID –– both directrices are curves.
CONOID –– one directrix is a curve while the other is a straight line.
WARPED PLANE –– both directrices are straight lines.

Surfaces.
Ruled surfaces with two Diretrices.
σ2 – plane director.
l1
l'1
m1
M1
l2
M2
m2
l'2
σ2
n2
n1
CYLINDROID
The surface called a cylindroid is generated by a moving straight line (generatrix) which in all its positions remains parallel to a given plane (called "the plane director“ or “plane of parallelism”) and intersects two curved guide lines (two directrices).
If the directrices are plane curves, then, of course, they must lie in different planes.

Surfaces.
Ruled surfaces with two Diretrices.
σ2 - plane director.
х
m2
n1
m1
σ2
σ2
n2
M2
M1
CONOID
The surface called a conoid is generated by a moving straight line (generatrix) which all the time remains parallel to a given plane (called the plane director or “plane of parallelism”) and intersects two directrices one of which is a curve, the other being a straight line.
If the curve is a plane one, then it must not lie in the same plane with the second directrix which is a straight line.

Surfaces.
Ruled surfaces with two Diretrices.
σ1 - plane director.
σ1
х
σ1
m2
M1
n2
n1
m1
M2
The surface of a warped plane (hyperbolic paraboloid) is determined by a plane director and two noncoplanar (skew) straight-line directrices. A straight-line generatrix moving along the directrices (and remaining parallel to the plane director) describes the surface of a hyperbolic paraboloid.
It can also be obtained by planar parallel motion of one parabola as generatrix along the other parabola as directrix.
WARPED PLANE (hyperbolic paraboloid)

Surfaces.
Ruled surfaces with two Diretrices.
WARPED PLANE –– both directrices are straight lines.
x
m2
n2
n1
m1
m, n – directrices ,
П1 – plane director.
l12
l11
M2
M1

Surfaces.
Ruled surfaces with two Diretrices.
Right Helicoid
Right Helicoid — ruled surface with a plane director perpendicular to one directrix — straight line (axis of helicoid) while the other directrix is a helix. Straight line directrix and axis of helix base cylinder are collinear (coinciding) lines.
Thus Right helicoid surface is conoid.
Algorithm: how to construct Right Helicoid.
Source data:  – directrix – straight line ⊥ plane director (P1), m – other directrix – helix, ℓ - generatrix.
Construct two views of helix – sinusoid and circle.
Construct a few intermediate locations of generatrix ℓ which intersect both directrices and remain parallel to the plane director P1.
Multiple locations of generatrix form the helicoid surface .
Point A belongs to the helicoidal surface. Missing projection can be found by constructing proper projections of generatrix.
i1
i2
m2
A2
l2
m1
A1
l1

Surfaces.
Ruled surfaces with two Diretrices.
Skew Helicoid
Skew Helicoid – ruled surface where generatrix ℓ moves along two directrices (one is m – helix, the other i – its axis) and remains parallel to the base cone of revolution (constant angle between axis and generatrix of a cone).
Curve of intersection between skew helicoid surface and a plane, perpendicular to its axis is called Archimedean spiral.
Point A belongs to the skew helicoidal surface. Missing projection can be found by constructing proper projections of generatrix.
i1
i2
m2
A2
l2
m1
A1
l1

Surfaces.
Ruled surfaces with two Diretrices.
Right Helicoid
Skew Helicoid

Surfaces.
Ruled surfaces with three Diretrices.
1 – general view
(3 directrices – curved lines)
2 – double-skew cylindroid
(2 directrices – curved lines)
3 – double-skew conoid
(2 directrices – straight lines)
4 – one-sheet hyperboloid
(3 directrices – straight lines)
Generally Ruled Surface unambiguously represented by moving generatrix along three directrices.

Surfaces.
Ruled surfaces with three Diretrices.
One-sheet Hyperboloid — can be formed by moving straight line generatrix along three skew lines (straight line directrices) which are not parallel to any plane. Generatrices can be directrices and vice versa. Outline — hyperbola.

Surfaces.
Ruled surfaces with three Diretrices.
Skew Wedge surface (type of double-skew cylindroid surface) — two directrices are smooth curves and one is a straight line. All directrices are lying in parallel planes. On construction Chords are divided proportionally.
А1
11
D1
В1
С1
11
А2
12
D2
В2
С2
12
n12
n22
n32
n31
n21
n11

Channel Surfaces.

Channel Surfaces.
On
Ln
L1
m

Channel Surfaces.
A1
A
m1
O1
l
m2
l2
l1
O2
A2
O
m

Channel Surfaces.

Channel Surfaces.
Screw Torus

Surfaces.
Positional problems.
Intersection between a Line and a Surface.
1
11
21
2
12
22
T1
11=21
22
12
S1
S2
a1
=B1
K1
B2
T2
S1
K2
a2
S2
3
Intersection between 5-faced prismatic horizontally perpendicular surface and a straight line.
Segment of a frontally perpendicular cylindrical surface intersects with a straight line
Horizontally projecting straight line intersects:
(а) - pyramidal surface
(b) – conical surface

Surfaces.
Positional problems.
Intersection between a Line and a Surface.
M1
M2
A1
b2
A2
R1
K1
B1
T1
11
42
32
21
41
N1
31
N2
K2
12
22
T2
R2
B2
x
5
b1
Intersection between АВ-line and cylindrical surface.
K – piercing point.

Intersection between АВ-line and surface of revolution.
Surfaces.
Positional problems.
Intersection between a Line and a Surface.
6
a1
11
a2
K'1
1'2
b1
b2
5'1
21
31
41
51
4'1
3'1
2'1
1'1
K1
K'2
K1
2'2
3'2
4'2
5'2
12
22
32
42
52

Surfaces.
Positional problems.
Intersection between a Line and a Surface.
8
12
N2
11
x
M1
N1
O1
M2
21
22
x'
2'1
M'1
1'1
r
O2
N'1
7
N1
141
132
112
91
l1
51
71
11
M1
M2
32
52
111
72
92
l2
12
61
21
41
81
101
122
121
142
82
62
42
22
131
31
N2
102
O'1
Projecting surfaces
Intersection between a sphere and oblique line

Surfaces.
Positional problems.
Intersection between a Line and a Surface.
10
a2
k2
a1
k1
h2
h1
b1
b2
m2
n2
m1
n1
h
9

Surfaces.
Positional problems.
Intersection between a Line and a Surface.
n1
12
m1
11
a1
21
k1
b1
h1
S1
h2
b2
k2
a2
n2
m2
S2
11

Surfaces.
Positional problems.
Intersection between surfaces.
Positional problems determine relative position and mutual belongings of objects and can be solved on their orthographical views. Basic positional problem: construct curve of intersection between two surfaces.
If 1–2 surfaces are perpendicular to principal planes (projecting surfaces), then positional problems can be solved without additional cutting surfaces.

Intersection between Surfaces
A2
A1
B2
C2
C1
B1
A3
=C3
B3
S2
S3
S1
12
32
(5)2
(7)2
(8)2
13
33
53
83
(7)3
(6)3
43
(2)3
41
11
31
21
61
81
51
71
22
42
(6)2
Surface perpendicular to a principal plane

Intersection between Surfaces.
Y34
12
(2')2
(3')2
42
(5')2
(4')2
32
22
(1')2
52
11
41
(2)1
(3)1
(3')1
4'1
51
(2')1
5'1
1'1
Y1'
Y6
62
61
6'1
(6')2
Y6
Y1'
Y34
Both Surfaces are perpendicular to principal planes
Y1
13
33
23
43
53
63
1'3
3'3
2'3
6'3
5'3
4'3
Circle
Ellipse
Straight Line

МТК
Intersection between Surfaces.
Polyhedrons. Both Surfaces are perpendicular to principal planes

МТК
Intersection between Surfaces.
f2
h1
62
B1
22
32
42
52
72
f1
h2
11
A1
21
31
41
51
71
81
91
101
61
12
82
102
A2
B2
92

Intersection between Surfaces.
B2
A2
C2
C1
B1
A1
11
21
31
41
51
61
S1
42
32
(6)2
12
(5)2
22
S2
S'2
П'2
6'2
4'2
5'2
3'2
1'2
2'2
K'2
L'2
M'2
N'2
O'2
P'2
B'2
A'2=C'2
K1
L1
N1
M1
P1
O1
L2
N2
P2
O2
M2
K2
h2
h1
f2
f1

Intersection between Surfaces. Cutting Surface Method.
To construct curve of intersection between surfaces we can use
Cutting Surface method.
As cutting surfaces we can use Cutting Planes or Cutting Spheres.
Projection of a Curve of intersection between surfaces lies in an area of intersection between projections of outlines of surfaces.
To construct curve of intersection it’s desirable to indicate characteristic points: — on the outline (upper, lower, left, right).
Ф2
n
Ф1
m
M
N

Intersection between Surfaces.
Cutting Plane Method.
S2
(2')2
(3')2
42
(5')2
(4')2
32
22
52
(4')3
(4)3
63
6'3
(3)3
23
2'3
1'3
11
1'1
21
31
3'1
2'1
41
4'1
61
6'1
S3
S1
Algorithm
Determine area where curve of intersection can exist.
Determine which surfaces of a model are perpendicular to principal planes.
Define base points for curve of intersection.
Assign type of cutting surfaces.
Define maximum number of cutting surfaces and their range.
Construct points of a curve of intersection.
Define base points where visibility of a curve changes.
Join all obtained points by a curve line in a proper sequence.
(3')3
13
hyperbola
ellipse
circle
62
(6')2
53
5'3
12
(1')2
5'1
51
Y3
Y3

Intersection between Surfaces.

Intersection between Surfaces.
Cutting Plane Method.
RC1
RC2
RК1
RК2
RC3

Intersection between Surfaces.
Τ1
Γ2
21
11
42
32
Σ1
51
61
52
53
63
42
62
12
22
32

Intersection between Surfaces.
Cutting Plane Method.
а1
61
81
41
42
92
31
Σ1
Δ1=f1
72
52
22
b2
h2
12
f2
51
61
A1
71
h'2
h''2
а2
B1
D1
S1
21
h'1
h'2
b1
32
81
A2
B2
C2
11
Common symmetry plane
S2
91
C1
D2

Intersection between Surfaces.
Cutting Sphere Method.
Coaxial surfaces of revolution (i.e. surfaces with a common axis)
intersect along circles.

Intersection between Surfaces.
Cutting Sphere Method.
Cutting surfaces - spheres
Theorem:
Two coaxial surfaces of revolution intersect at circles which lie in planes, perpendicular to the axis of revolution, and centers of the circles lie on that axis.
If center of a sphere lies on the axes of revolution, then sphere intersects surface of revolution at circles.
Application of the method:
1. Both surfaces are surfaces of revolution;
2. Axes of these surfaces are intersecting lines;
3. A plane, formed by intersecting axes, should be parallel to a principal plane.
а.
c.
b.
d.
A
B
A=B

Intersection between Surfaces.
Cutting Sphere Method.
Algorithm how to construct curves of intersection between two surfaces of revolution (cones).
1. Determine point of intersection between axes of revolution as a center of cutting spheres (О2 point).
2. Find projections of base points for curves of intersection.
3. Define band of radii R for cutting spheres: RMAX and RMIN. RMAX – distance from the center of a sphere to the outermost point, which belongs to the curve of intersection. RMIN – radius of a sphere, which is tangent to the one surface of revolution and intersects the other one.
4. Construct sphere of a radius R: RMIN
5. Construct multiple of cutting spheres for different R. Join gained points and obtain curve of intersection between two surfaces.
Cutting surfaces – concentric spheres
O2
Rmin
Rmax

Intersection between Surfaces.
Cutting Sphere Method.
Cutting surfaces – eccentric spheres
Algorithm how to construct curves of intersection between two surfaces of revolution
(a cone and a torus).
Application of the method: both surfaces of revolution have a common plane of symmetry.
1. Determine base points 12, 22.
2. Construct plane ∑2 (sigma), which includes axes of a torus t2. It intersects the torus at a circle of R-radius (radius of tube).
3. On the axis of a cone find center of a sphere, which intersects torus at that circle - point О2.
4. Draw a sphere of R’-radius and О2 – center.
5. Find a circle of intersection between a sphere and a cone. (on the plane П2 projects to a line).
6. Find points of intersection between two circles – points 3(4).
7. Through the axis t2 draw new cutting plane ∑'2 and repeat constructions.
8. Join points 1, 3, 5, 2 by a smooth curve.
R’'
O'2
O2
R’
t2
32
12
52
22
Σ2
Σ '2
=(4)2
=(6)2
R

Intersection between Surfaces.
Cutting Sphere Method.
Curve of intersection
Curve of intersection
Concentric spheres
Eccentric spheres

Intersection between Surfaces.
Particular case.
In general two second-order surfaces of revolution intersect at a four-degree curve. But in some cases the curve of intersection decomposes into two planar second-degree curves. It happens when both intersecting surfaces of revolution (a cylinder and a cone, two cones, an ellipsoid and a cone, etc.) are circumscribed about a common sphere.

Intersection between Surfaces.
Cutting sphere method. Particular case.
Two second-order surfaces circumscribed about a third second-order surface intersect each other along two second-order curves.

Intersection between Surfaces.
Cutting sphere method. Particular case.
Two second-order surfaces circumscribed about a third second-order surface intersect each other along two second-order curves.

Intersection between Surfaces.
Cutting sphere method. Particular case.
Curve of intersection
Curve of intersection -
elliptical arcs

Developments.
Properties of developments.
Each point on the development corresponds to a single point of a surface.
Straight lines of a surface remain straight on a development.
Straight line segments preserve their lengths.
An angle formed by lines on a surface remains equal to an angle between the corresponding lines on the development.
The area of a closed domain on a surface retains its magnitude within the corresponding closed domain on the development.
Shortest distance (beeline) on a surface develops into a straight line on a development (developed distance).
Parallel lines develop to parallel lines.
Some curved surfaces can be developed so that they coincide completely (with all their points) with a plane without stretching or shrinking. Obtained planar figure is called development of a surface while surfaces are called developable.
They comprise polyhedrons and ruled surfaces where adjacent rectilinear generatrices are parallel or intersect, or are tangent to sphere curve.
All double-curved surfaces and the ruled surfaces which cannot be developed into a plane are called nondevelopable (or warped) surfaces.

Methods of exact development.
1. Triangulation method.
2. Radial-line method.
3. Stretch-out-line (right section) method.
Developments.
Types of developments:
Exact.
Approximate.
Conventional.
Exact developments.
Developments of polyhedrons, right circular cylinder and cone can be constructed theoretically exactly. Cylinder develops to a rectangle, cone – to a circular sector.
π D
a
L

Developments.
Triangulation method.
Development of a pyramid. Lateral faces of a pyramid are triangles. To find true sizes of triangles you can use transformation of a drawing or find true lengths of corresponding edges. Because base of a pyramid is parallel to П1-plane, it’s sufficient to find true lengths of lateral edges AS, BS, CS. After that construct, for example, ВCS-face, and supplement construction adding ASB and АSC faces.
S2
C2
B2
A2
B'2
C'2
A'2
S1
C1
B1
A1
A'1
B'1
C'1
S0
C0
B0
A0
A0
B0
RSA
RSB
RSС
RAB
RСB
RBC
RBA
RCA

Developments.
Radial Line method.
C0
A1
A2
C0
A0
B0
D0
D1
C1
F1
B1
C2
B2
E1
F0
F2
D2
E2
E0
D0
This method is used when a base of a figure is parallel to the one principal plane of projection and generatrices are parallel to the other principal plane.
Find true size of А2D2F0C0–face revolving it around frontal А2D2. Point F2 moves perpendicularly to А2D2 to location F0, which can be found by protracting А1В1 – line segment (the true size of АВ) from the point А2.
From the point B2 draw perpendicular to А2D2 and find B0, protracting it from the point C0 by В1С1-line segment, etc. Thus we obtained row of points А0≡А2, В0, С0 ..., which define a margin for developed bottom base of a figure.

Developments.
Stretch out Line method.
This method is used when generatrices are parallel to a principal plane of projection.
Cut ABCDEF-prism by the plane γ, perpendicular to the lateral edges of prism. Construct section of the prism by the plane in question – Δ123. Define true lengths of sides of 123-triangle. On a free space of a drawing draw horizontal straight line a. From the arbitrary point 10 on this straight line protract line segments [1020], [2030], [3010], equal to the sides of 123-triangle. Through the points 10, 20, 30, 10 draw straight lines, perpendicular to the a-line, and protract from the points 10, 20, 30, 10 line segments, equal to lengths of corresponding lateral edges ([12A2], [12D2], [22B2], [22E2], [32С2], [32F2]), Then join obtained points A0B0C0A0 and D0E0F0D0 line segments. Obtained planar figure A0B0C0A0D0E0F0D0 is a development of a lateral surface of the prism.
To construct full development of a prism supplement obtained development by true figures of top and bottom bases.
Construct development of ABCDEF - prism.
A1
C1
D1
F1
A2
B2
12
22
A'2
C'2
B'2
A0
B0
C0
A0
11
D0
F0
E0
D0
20
10
30
10
D'2
E'2
22
32
12
E2
F2
D2
D0
A0
B1
E1
C2
32
Ƴ
21
31
F'2

Development of a conical surface. Triangulation method.
Substitute (approximate) the conical surface by polyhedral pyramidal surface.
Define true lengths of lateral edges (method of revolution is used).
Construct true sizes of faces.
Obtained points of cone base join by a smooth curve.
Size and number of straight line segments which approximate a curve directrix of a cone depend on curvature of a curve and size of a cone.
Approximate developments.
11
S2
S1
21
31
51
61
71
81
1'2
1'1
6'1
5'1
7'1
41
4'1
8'1
3'1
2'1
6'2
5'2
7'2
4'2
8'2
3'2
2'2
2'2
3'2
4'2
5'2
6'2
7'2
8'2
1'2
r=|1121|
R=|S222|

Conventional developments.
Development of Toroidal surface— Zone Method
Each parallel of a surface is spaced an equal distance, m, apart along the surface. Cones are passed through the surface so that they pass through two parallels at the outer surface of the toroidal. The largest cone with element R1 is found by extending it through where the equator and the next parallel intersect on the surface in the front view until R1 intersects the extended centerline of the toroidal surface. Elements R2 and R3 are found by repeating this process.
The development is begun by laying out the largest zone, using R1 as the radius, on the arc that represents the base of an imaginary cone. The breadth of the zone is found by laying off distance m from the front view to the development and drawing the upper portion of the zone with a radius equal to R1-m using the same center.
The next zone is drawn using radius R2 with its center located on a line through the center of arc R1. The last zone will appear as a circle with R3 as its radius.
The lengths of the arcs can be established by dividing the top view with vertical cutting planes that radiate through the poles - meridians.
Preliminary nondevelopable surface is approximated by segments of developable surfaces. Then construct development of every segment, and the whole totality of developments gives conventional development of nondevelopable surface.
b
c