Transformation of a Drawing

Transformation of a Drawing Replacing of planes of projection
(projecting to auxiliary planes).
In this method location of an object doesn’t change while one of planes of projection is replaced by a new one, perpendicular to the other plane of projection.
Properties of transformation:
New projectors perpendicular to a new axes;
Distance between a new projection and new axes is equal to a distance between replaced (old) projection and replaced axes.

Transformation of a Drawing. Replacing of planes of projection
(projecting to auxiliary planes)
Projecting of a point on auxiliary plane. Point А is given by its projections in П2/П1 planes. Construct projections of a point A, if the plane П2 is replaced by a new vertical plane П'2.
Horizontal projection А1 doesn’t change because location of П1 is not changed. To determine new frontal projection А'2 draw perpendicular prom А1 point to intersection with П'2 plane. А'2 – new frontal projection of А point.
А'2АХ1 = А2АХ. А2АХ is Z-coordinate of a point А (ZА). It means that if frontal plane of projection is replaced, Z-coordinates of entire figure doesn’t change.
Algorithm:
Draw perpendicular (new projector) from a point А1 to a new axes of projections X1.
On a newly obtained projector protract ZA-coordinate of a point А from the point АХ1,
i.е. А′2АХ1 = А2АХ.
A
x
x
A2
П2
П1
Ax
A1
A2
Ax
A1
П1
П2
x1
П'2

Transformation of a Drawing. Replacing of planes of projection
(projecting to auxiliary planes)
Application of the method.
Oblique line segment transforms to a line segment parallel to the plane of projection and in succession perpendicular to the plane of projection.
Algorithm :
1. Implement new frontal plane (primary auxiliary plane) П′2 ІІ [АВ] to transform А1В1 to a line parallel to a principal plane.
2. Construct frontal projection of [А′2 В′2]. Z-coordinates of points А and В doesn’t change.
3. Implement new horizontal plane (secondary auxiliary plane) П′1 ⊥ [А′2 В′2].
4. Construct horizontal projection of a line [А′1 В′1] in a new system П′1.
x
x2
П1
П2
П'2
A2
B2
A
B
B'2
B1
A1
A'2

Transformation of a Drawing. Replacing of planes of projection
(projecting to auxiliary planes)
Construct a true size of a plane segment АВС.
Algorithm :
1. Draw horizontal line in the plane (АВС) – h2.
2. Construct horizontal projection of a horizontal line – h1.
3. To implement a primary auxiliary plane П’2 draw a new axes x’2 perpendicular to h1.
4. Construct new frontal projection of АВС-plane – A’2B’2C’2 in a plane П’2 .
5. To implement a secondary auxiliary plane П’1 draw a new axes x’’1 parallel to A’2B’2C’2.
6. Construct new horizontal projection of АВС-plane – A’1B’1C’1 in a plane П’1.
A’1B’1C’1 is a true size of АВС-plane.
x
A2
B2
C2
Cx
Bx
Ax
B1
A1
C1
TS

Transformation of a Drawing. Revolution about an axes
perpendicular to a plane of projection
In this method location of planes of projection doesn’t change while an object revolves about an axes, perpendicular to one plane of projection.
Revolution about an axes ⊥ П2.
All points of a figure are rotated at the same angle α.
α
M'
α
O
M
R
O2
M2
M'2
x
П2
П1
M1
R
x
O2
α
M1
M'1
O1
M2
M'2
90°
Radius of revolution to a TL
Frontal principal plane perpendicular to an axes of revolution.

Transformation of a Drawing. Revolution about an axes
perpendicular to a plane of projection
Revolution about an axes ⊥ П1.
O1
R
M1
O2
M2
90°
M'2
M'1
TL of a Line segment
Horizontal principal plane perpendicular to an axes of revolution.
x
M
П
1
O1
O
R
α
П2
M2
M'2
M'
M'1
M1

Transformation of a Drawing. Revolution about an axes
perpendicular to a plane of projection
B2
A2
C2
B1
A1
C1
A
K'2=
How to find a True Size of a Surface Segment
Algorithm:
1. Construct horizontal line h in the plane ABC.
2. Revolve horizontal projection of ABC-triangle to locate h1 ⊥ П2.
3. Construct frontal view of ABC-triangle. It projects to a line perpendicular to the frontal principal plane.
4. Revolve obtained view to locate it parallel to horizontal principal plane. A”B”C”⎟⎟ to x-axes .
5. Obtained horizontal view is a True Size of a ∆ABC.

Transformation of a Drawing. Revolution about an axes perpendicular to a plane of projection without indicating an axis in a drawing.
(Planar Parallel Motion)
How to find a True Size of a Surface Segment
Algorithm:
1. Construct horizontal line h in the plane ABC
2. Move horizontal projection of ABC-triangle to locate h1 ⊥ П2.
3. Construct frontal view of ABC-triangle. It projects to a line perpendicular to the frontal principal plane.
4. Move obtained view to locate it parallel to horizontal principal plane. A”B”C”⎟⎟ to x-axes .
5. Obtained horizontal view is a True Size of a ∆ABC.
B2
A2
C2
B1
A1
C1
K'2 =
In this method location of planes of projection doesn’t change while each point of an object moves in a plane, parallel to one plane of projection.

Transformation of a Drawing. Revolution about an axes perpendicular to a plane of projection without indicating an axis in a drawing
(Planar Parallel Motion)
Find a True Size of a dihedral angle between plane segments.
Algorithm:
Move ABCD- figure parallel to П2 to locate ВС-edge parallel to horizontal principal plane П1 - В′2C′2⎟⎟ x-axes.
Construct horizontal view of A’B’C’D’-figure. Each vertex of a figure moves in a plane, parallel to П2 (In П1 parallel to х-axes). Vertices А’1, В’1, C′1, D′1 are placed on intersections between horizontal and vertical projectors.
Move obtained A’B’C’D’-figure parallel to П1 plane to locate В’С’-edge perpendicular to a frontal principal plane П2 - В”1C”1 ⊥ x-axes.
Construct frontal view of A”B’’C’’D’’-figure. Vertices А’’2, В’’2, C’′2, D’′2 are placed on intersections between horizontal and vertical projectors.
A”B’’C’’ and B’’C’’D’’-planes are perpendicular to the frontal principal plane, because common
edge В” C”⊥ П2. It enables to find a true size of dihedral angle between them on the frontal view.
B2
D2
A2
A1
B1
D1
С2
С1

Transformation of a Drawing. Replacing of planes of projection
(projecting to auxiliary planes)
Polyhedrons. True size of a planar section.
Algorithm:
1. Construct primary auxiliary plane parallel to the section to project section to a true size.
2. On projecting a section to primary auxiliary horizontal plane Y-coordinates of a section don’t change, and they are protracted from a new X1 axes on the proper projectors, perpendicular to the X1 axes.
True Size of a polyhedron planar section is obtained.
Section can be constructed in a free space of a drawing by placing a new axes parallel to a section plane and protracting Y-coordinates of points 1, 2, 3, 4, 5 and 6 perpendicular to the new axes.
S1
A1
E1
F2
C1
C2
D1
S2
A2
B2
B1
F1

Transformation of a Drawing. Replacing of planes of projection
(projecting to auxiliary planes)
Find a distance between two skew lines.
B2
D2
A2
A1
B1
D1
С2
С1
П'2
x'
П1
C'2
A'2
D'2
B'2
x
П1
П2
x''
П'1
П'2
Distance between lines
C''1
=D''1
A''1
B''1
Distance between a line and a point

Transformation of a Drawing. Replacing of planes of projection
(projecting to auxiliary planes)
Find a distance between a point and a plane.
B2
D2
A2
A1
B1
D1
С2
С1
D'2
C'2
A'2
B'2
x
П1
П2
П'2
x'
П1
h2
12
h1
11
Distance between a point and a plane

Intersection between planes

Intersection between a Line and a Plane
A Line and a Plane have one common Point

Intersection between a Line and a Plane
A Line and a Plane have one common Point
Algorithm.
1. Construct cutting plane which would contain the given line.
2. Find line of intersection between the given plane and cutting plane.
3. Determine point of intersection between obtained line of intersection
and given line.
How to find a piercing point
π1
X
π2
M2
M1

Intersection between two oblique planes.
Cutting plane method

Intersection between two oblique planes.
Cutting plane method
Line of intersection between two planes is a multiple of points which simultaneously belong to both planes. To find a line of intersection between two planes it’s sufficient to find two points which belong to both planes.
Two Cutting Planes are used to find two common points.

Intersection between two oblique planes.
Cutting plane method
Algorithm:
1. Construct two cutting planes intersecting two given oblique planes and perpendicular to some principal plane.
2. Construct lines of intersection between two cutting planes and two oblique planes.
3. Determine points of intersection between obtained lines.
4. Construct line of intersection joining determined points.

Axonometric projection
The word 'axonometry' (Greek) consists of two words: 'axon'—axis and 'metreo'—I measure, and means 'measuring with the aid of axes', or 'measuring along axes'.
The method of axonometric projection consists in the fact that an object is referred to some coordinate system and then is projected by parallel lines or rays onto a plane together with the system of coordinates.
Axonometric projections differ from orthographic (orthogonal) projections in that in axonometry an object is projected only onto one plane of projection called the axonometric (or pictorial) plane and is placed in front of the picture plane so as to expose three sides to the viewer.
In mechanical engineering axonometric projections are used as an auxiliary to orthographic projections of a mechanical part when it is the necessary to give a clearer picture of its shapes which are difficult to imagine from the orthographic projections. Without the axonometric picture it is sometimes very difficult to visualize the shape of the object from the three orthographic projections alone.
The basic statement of axonometry" was formulated by K. Polke (in 1851) in the form of the following theorem: any three line segments emanating from a single point in the plane may be taken as a parallel projection of three equal and mutually perpendicular line segments in space. In the sixties of the 19-th century G. Schwartz generalized Polke's theorem. He proved that any complete quadrilateral in the plane may always be regarded as a parallel projection of a tetrahedron similar to any given one.
Therefore, any three non-coincident straight lines passing through a point in the plane may be taken for the axonometric axes with various scales.

Axonometric projection
A1
x
AX
A
O
z
y

Axonometric projection
A line segment e is laid off on the axes x, y, z, and is taken as a unit of measurement along these axes (the true unit). The line segments ex, ey, ez on the axonometric axes represent the projections of the segment e. In the general case they are not equal to /, and are not equal to one another. The segments ex, ey, ez are called the axonometric units and are used for measuring along the axonometric axes
The ratios
KX= ; KZ= ; KY=
are called distortion factors along the axonometric axes.
Regarding to distortion factors axonometric views can be:
trimetric – all distortion factors are different (KX ≠ KY ≠ KZ);
dimetric – any two distortion factors are equal (for example KX = KY);
isometric – all distortion factors are equal to each other(KX = KY = KZ).

Axonometric projection
Isometric projection

Axonometric projection
Dimetric projection

1. From the point О′ along О′Х′ - axes protract distance XA = |ОAx| .
2. From the obtained point A′x draw a straight line, parallel to О′Y′.
3. On the obtained straight line protract distance YA= |AxA1| and obtain А′1 point.
4. From the point А′1 draw a straight line, parallel to O′Z′ axes. On the obtained straight line protract distance ZA= |AxA2|.
5. Point А′ is a projection of a point А in axonometric coordinate system.
Algorithm how to construct axonometry by the orthographic view
Axonometric projection
Axonometric view
Orthographic view
x
y
O
A2
z
O'
A1
XA
ZA
YA
Ax
X'
z'
y'

Axonometric projection
Isometric projection
Frontal Dimetric projection
A=1.22d – major axes
CD=0.7d – minor axes
d – diameter of a circle
Major axes of an oval is always perpendicular to the axonometric axes, which doesn’t belong to the plane of a circle

Axonometric projection
Algorithm:
1. In axonometric coordinate system X’, Y’, Z’ protract points 1, 8, 2, 3.
2. Draw line segments |45|, |67| || to X’ axes.
3. Join points 1, 4, 3, 5, 8, 7, 2, 6 and obtain axonometry of the base.
4. Construct lateral edges – from vertices of the base draw straight line segments || to Z-axes at a distance of polyhedron height ().
5. Join end points of obtained straight line segments.
6. Construct (if necessary) axonometric projection of М point, which belongs to a lateral face of a polyhedron.
How to construct axonometric projection of polyhedron
O'
X'
z'
y'
y
z
O
x
1
4
3
5
8
7
2
6
M1
M2
0

Axonometric projection
Algorithm how to construct orthographic view by the axonometry
Given data: cone of revolution in axonometric projection. Point K belongs to a surface af the cone.
1. Through a point К draw a straight line – generatrix of a cone.
2. Draw its projection on xy-plane.
3. Construct point К'' on the obtained projection.
4. Find coordinates ХK and YK of К'‘-point.
5. Draw obtained coordinates on the orthographical view.
O
x
y

Orthogonal frontal isometric projection

Orthogonal frontal dimetric projection
D – true diameter of a circle

Nonorthogonal frontal isometric projection – cavalier (military) pictorial view

Nonorthogonal horizontal isometric projection