Lecture_4-Slides_(Part_1).pptx

Lecture 3: Geometric Construction
❑ Geometrical Constructions
❑ Construction Geometry
❑ The Ruler & Compass Geometry
FROM PAGE
CHAPTER 5
OF THE
PRESCRIBED
BOOK

Learning Outcomes
Geometric Construction Nomenclature
Geometric Construction Principles
Polygon Construction Principles
Constructing Tangent Arcs
Conic Sections

■ The accurate construction or drawing of basic drawing
primitives/entities (such as lines, angles, and shapes)
primarily USING A RULER AND A COMPASS ONLY
■ The relationship between the basic drawing primitives/entities
(e.g., lines, angles, and shapes)
■ Recognizing the geometry that exists within and between
objects to enable creation of solid models or multi-view
drawings
Geometric Construction is about:

Discussion Points
Points and Lines
Cartesian Coordinate System
Planes
Polygons
Basic geometric construction principles, e.g.,
bisection of Lines & Angles, etc.
Arcs & Tangency

TERMINOLOGIES #
POINTS
■ Points indicate exact locations in space and are represented
by a CROSS (and NOT A DOT) at the point of exact
location.
■ A point located on a line is represented by a short dash.
■ Points are considered dimensionless (that is, they have no
Height, Width or Depth)
■ Mathematically, points are defined on the Cartesian plane
by a set of (x, y) coordinates.

ABSOLUTE
COORDINATE
S
RELATIVE OR
INCREMENTAL
COORDINATES
TWO
SYSTEMS
Under ABSOLUTE the
(X, Y) values of a point
reference the ORIGIN. Under RELATIVE a
point is defined by
referencing the
previous point.
SYSTEM
FOR POINTS

X
Y
0 1 2 3 4 5 6 7 8 9
3
2
1
0
6
5
4
A = X3, Y2
B = X4, Y4
C = X7, Y1
D = X8, Y5
A
B
D
C
SYSTEM
FOR POINTS

X
Y
0 1 2 3 4 5 6 7 8 9
3
2
1
0
6
5
4
A
B
D
C
A ref zero= X3, Y2
B ref A = X1, Y2
C ref B= X3, Y-3
D ref C= X1, Y4
COORDINATES SYSTEM
for POINTS

■ LINE: the locus of a point between two or more
locations
■ A STRAIGHT LINE is the shortest distance between
two points.
■ Lines are considered to have length, but no other
dimension such as width or thickness.
■ Lines come in different forms
TERMINOLOGIES #
LINES

■ Parallel lines
■ Two or more lines that are always the same distance
apart
■ Perpendicular lines
■ Two lines that are at a 90° angle
TERMINOLOGIES #
LINES

X
Y
0 1 2 3 4 5 6 7 8 9
3
2
1
0
6
5
4
A
D
B
C
Line AD
Line BC
SYSTEM
for LINES

X
Y
0 1 2 3 4 5 6 7 8 9
3
2
1
0
6
5
4
A
D
0°
90°
180°
270°
45°
❑ The polar system is relative system that references the previous
point and locates the next point by distance and an angle
Point “D” uses point “A” as the reference
point.
The polar coordinate of “D” is: 5.0 < 45°.
5.0 is its LENGTH,
45° is its ORIENTATION in the coordinate
plane.
for
LINES

POINTS ABSOLUTE RELATIVE POLAR
1 (10,10) 10,10 (10,10)
2 (30,10) @20,0 @20<0o
3 (30,20) @0,10 @10<90o
4 (10,20) @-20,0 @-20<180o
5 (10,10) @0,-10 @-10<270o
SYSTEM
in AutoCAD

■ An ANGLE defines relative orientation of
two lines with respect to each other,
■ VERTEX
■ a Point at which two lines intersect
Vertex
TERMINOLOGIES #ANGLES
&VERTICES

TERMINOLOGIES #ANGLES
&VERTICES
■ Acute Angle
■ Measures less than 90°
■ Obtuse Angle
■ Measures more than 90°
■ Right Angle
■ Measures exactly 90°

■ Degrees (°)
■ e.g. 48°
■ Minutes (')
■ e.g. (48° 18')
■ Seconds (”)
■ e.g. (48°18' 27”)
TERMINOLOGIES #UNITS OF
ANGLES

#CONVERTING ANGLES FROM ONE
UNIT TO ANOTHER

#CONVERTING ANGLES FROM ONE
UNIT TO ANOTHER
To flip 48.30750 back to degrees, minutes, and
seconds
▪ 48.30750 = 480 + (0.3075 x 60 = 18.45)Ꞌ
▪ 480 (18.45)Ꞌ = 480 18Ꞌ + (0.45 x 60)ꞋꞋ
▪ 480 18Ꞌ 27ꞋꞋ

■ Planes are defined by:
▪Three points not lying in a straight line
▪Two parallel lines
▪Two intersecting lines
▪A point and a line
4. BASIC GEOMETRIC ENTITIES
#PLANES

Planes are defined by the axes that lie on the plane.
Y
Z X
The XZ Plane
X
Y
Z
The XY Plane.
Y
X
Z
The YZ Plane
# CARTESIAN COORDINATES SYSTEM
#of
PLANES

■ Radius
■ Distance from the
center of a circle to its
edge
■ Diameter
■ Distance across a circle
through its center
■ Circumference
■ Distance around the
edge of a circle
■ Chord
■ Line across a circle
that does not pass at
the circle’s center
#CIRCLES
■ Has 360°
■ Quadrant
■ One fourth (quarter)
of a circle
■ Measures 90°
■ Concentric
■ Two or more circles
of different sizes that
share the same
center point

Defined as a closed plane figure with three or
more straight lines:
❑ Triangles
❑ Quadrilaterals
❑ Hexagons
❑ Octagons
#POLYGON

BASIC GEOMETRIC ENTITIES #BISECTING
LINES
1. Given a line AB
1. With the centre A and a radius
greater than half AB, draw arcs
on both sides of AB
1. With the centre B and the same
radius, draw arcs intersecting
the previous arcs at C and D
1. Draw a line joining C and D and
cutting AB at E
1. AE = EB = 1/2*AB
1. Note that CD bisects AB at a
right angle
A B
C
D
E
900

BASIC GEOMETRIC ENTITIES #BISECTING
ARCS
1. Given an arc AB
1. With the centre A and a radius
greater than half AB, draw arcs
on both sides of AB
1. With the centre B and the same
radius, draw arcs intersecting
the previous arcs at C and D
1. Draw a line joining C and D and
cutting AB at E
1. AE = EB = 1/2*AB
1. Note that CD passes through
centre O of arc
A B
C
D
E
O

B.G.E. #BISECTING ANGLES
1. Given two lines AB and BC
1. With B as centre and a
convenient radius, draw an arc
cutting AB at D and BC at E
1. With the centres D and E and
the same or any convenient
radius, draw arcs intersecting F
1. Draw a line joining B and F,
1. BF bisects the angle BAC, i.e.
ABF = FBC
A
C
B
D
E
F

B.G.E. #PERPENDICULARS TO GIVEN LINES
1. Given a line AB and a point P
near the MIDDLE of the line
1. With the P as centre and any
convenient radius R1 draw an
arc cutting AB at C and D
1. With radius R2 greater than R1
draw arcs intersecting each
other at O
1. Draw a line joining O and P and
cutting AB at E
1. Then PO is the required
perpendicular
A B
C D
P
900
O
R1 R1
R2 R2

1. Given a line AB and a point P
near the END of the line
1. With a point O selected in
space, draw an arc with a
radius equal OP, larger than a
semicircle, cutting AB at C
1. Draw a line joining C and O, and
produce it to cut the arc at Q
1. Draw a line joining P and Q
1. Then PQ is the required
perpendicular
O
B
C P
900
Q
A

1. Given a line AB and a
point P outside AB
2. With centre A and a
radius AP, draw an arc
EF, cutting AB or an
extension of AB at C
1. With centre C and a
radius equal to CP,
draw an arc cutting EF,
at D
1. draw a line joining D
and P, intersecting AB
at Q
1. Then PQ is the required
perpendicular
P
B
C
900
Q
A
F
E
D

B.G.E. # PARALLEL LINES GIVEN A POINT
1. Given a line AB and a
point P outside AB
2. With centre P and a
convenient radius R1,
draw an arc CD cutting
AB at E
1. With centre E and
same radius equal to
EP, draw an arc cutting
AB, at F
2. With centre E and
same radius equal to
FP, draw an arc cutting
CD, at Q
1. draw a straight line
joining P and Q, which
is required parallel line
P
B
C
Q
A F
E
D
R1

B.G.E. # DIVIDING LINES
1. Given a line AB draw another line
AC making an angle of less than 30
with AB
1. With the help of a compass, mark
7 equal parts of an suitable length
on line AC and mark them by
points 1’, 2’, 3’, 4’, 5’, 6’ and 7’.
1. Join the last point 7’ with point B
of the line AB
2. Now from each of the other
marked points 6’, 5’, 4’, 3’, 2’, and
1’, draw lines parallel to 7’B
cutting the line AB at 6, 5, 4, 3, 2,
and 1 respectively
3. Now the line AB has been divided
into 7 equal parts. You can verify
this by measuring the lengths
B
A
1 2
3
4 5
6 7
1’
2’
3’
4’
5’
6’
7’
C

B.G.E. # BISECT ANGLES
1. Given an angle ABC, with B as
centre and any radius draw an arc
cutting AC at D and BC at E
1. With D and E as centres and the
same or any convenient radius,
draw arcs intersecting each other
at F.
1. Join B and F. BF bisects the angle
ABC, i.e., ABF = FBC
C
B E
D F
A

B.G.E. # TRISECT A GIVEN RIGHT ANGLE
1. Given the right angle ABC, and any
radius, draw an arc cutting AB at D
and BC at E
1. With the same radius and D and E
as centres, draw arcs cutting the
arc DE at points Q and P.
2. Draw lines joining B with P and Q.
BP and BQ trisect the right angle
ABC. ABP = PBQ = QBC = 1/3 ABC
C
B E
D
A
P
Q

B.G.E. # DRAW A LINE AT ANGLE TO ANOTHER
1. Given PQ, and an angle AOB, with
centre O and any radius, draw an
arc cutting OA at C and OB at D
1. With the same radius and P as
centre, draw an arc EF cutting PQ
at F.
2. With F as centre and radius equal
to CD, draw an arc cutting the arc
EF at G.
1. from P draw a line passing through
G, to obtain the required line
B
O
D
C
A
Q
P F
G
E

B.G.E. # FIND THE CENTRE OF AN ARC
1. Given the arc AB, draw two chords
A and B of any length.
1. Draw perpendicular bisectors of
CD and EF intersecting each other
at O, then O is the required centre
B
O
D
A
C E
F

B.G.E. # DRAW AN ARC TOUCHING 2 STRAIGHT LINES GIVEN
THE RADIUS
1. Given the lines AB and AC, and a
radius, R, use the radius and the
centre A to draw two arcs cutting
AB at P and AC at Q.
1. With P and Q as centres and the
same radius, draw arcs
intersecting each other at O
1. With O as centre and radius equal
to R, draw the required arc.
B
P O
A
C
R
R
Q
R

Sharp corner
Fillet
Round
Round
B.G.E. #FILLETS AND ROUNDS

To draw the arc, we must find the location of the center of that arc.
How do we find the center of the arc?

To draw an arc of given radius tangent to two perpendicular lines
Given arc radius r
r
r

center of the arc
Starting point
Ending point
Given arc radius r

+
+
r
r
Given arc radius r

T.P.1
T.P.2
To draw an arc of given radius tangent to two lines
Given arc radius r
B.G.E. #ARCS AND TANGENCIES

C
To draw a line tangent to a circle at a point on the circle
Given

C
mark a tangent point
To draw a line tangent to a circle from a point outside the circle
Given

C1
C2
Tangent point
R1
R2
The center of two circles and tangent point must lie on the same
straight line !!!
A circle tangent to another circle

To draw a circle tangent to two circles I
+
C
2
Given
+
C
1
C
+
Example

+
+
C
1
C2
R +
R1
Given
Two circles and the radius of the third circle = R
R +
R2
R1
R
2
C
center of the arc
To draw a circle tangent to two circles I
R

C2
R2
When circle tangent to other circle
C1
Tangent point
R1
The center of two circles and tangent point must lie on the
same straight line !!!

Given
+
C
1
+
C
2
C +
To draw a circle tangent to two circles II
Example

+ +
C
1
C
2
R –
R2
To draw a circle tangent to two circles II
Given
R –
R1
R
1
R
2
C
R

+
C
1
+ C
2
To draw a circle tangent to two circles III
Given
R +
R2
R –
R1
R
1
R
2
C

Polygons can be:
#DRAWING POLYGONS
■ inscribed (drawn within a
circumference) or
■ circumscribed (drawn
around a circumference)

An inscribed polygon is
constructed for polygons
where the number of sides
and the distance across the
corners have been given.
6-sided polygon
Draw a 6-sided polygon, of a length x across corners.
• Draw a line AB equal to the given length, x
• With the centre A and radius AB draw a semi-circle BP
• Divide the semicircle into 6 equal parts ( 180° ÷ 6 = 30°
) which can be easily done by using the triangles (in this
case the 30° - 60° triangle). Number the divisions as 1, 2,
etc starting at P
• Draw a line joining A with point 2
• Draw a perpendicular bisector of line A2 and another
perpendicular bisector of line AB, and mark their
intersection as O (however in this particular example that
intersection is at point 4
• With centre O/4 and radius OA/4A, draw a circle
• With radius AB and starting from B, cut the circle at
points C, D,….2
• Draw lines BC, CD, etc thus completing the required
polygon
#INSCRIBED POLYGON
C
D
E

4-sided polygon
#INSCRIBED POLYGON
multi-sided polygons

A circumscribed polygon can
be constructed by determining
the number of sides and the
distance across the flats.
6-sided polygon
Draw a 6-sided polygon, of a length x across flats/sides.
• Draw a horizontal line AB equal to the given length, x,
and bisect it and mark the mid point O
• With the centre O and radius OA or OB draw a circle
• Divide the circle into 6 equal parts ( 360° ÷ 6 = 60° )
which can be easily done by using the triangles (in this
case the 30° - 60° triangle) to locate points C, D, E, & F.
• Draw tangents at points A, B, C, D, E, & F to intersect at
points 1, 2, 3, ….6.
• Draw a perpendicular bisector of line A2 and another
perpendicular bisector of line AB, and mark their
intersection as O (however in this particular example that
intersection is at point 4
• With centre O/4 and radius OA/4A, draw a circle
• With radius AB and starting from B, cut the circle at
points C, D,….2
• Draw lines BC, CD, etc thus completing the required
polygon
#CIRCUMSCRIBED POLYGON

6-sided polygon
#CIRCUMSCRIBED POLYGON
some feasible circumscribed polygons put together
8-sided polygon

Pentagon Hexagon Heptagon
Octagon Nonagon Decagon
5 SIDES 6 SIDES 7 SIDES
9 SIDES
8 SIDES 10 SIDES
#SOME REGULAR POLYGONS

■ Equilateral
■ All three sides are of equal length and all three angles are
equal
■ Isosceles
■ Two sides are of equal length
■ Scalene
■ Sides of three different lengths and angles with three
different values
■ Right Triangle
■ One of the angles equals 90°
■ Hypotenuse: side of a right triangle that is opposite the 90°
angle
#TRIANGLES

Quadrilaterals are figures with fours sides
■ Square
■ Four equal sides and all angles equal 90°
■ Rectangle
■ Two sides equal lengths and all angles equal 90°
■ Trapezoid
■ Only two sides are of equal length
■ Rhombus
■ All sides are equal length and opposite angles are equal
■ Rhomboid
■ Opposite sides are equal length and opposite angles are equal
■ Trapezium – No sides parallel
#QUADRILATERALS

Construction Principles
• Bisecting a Line
1. Draw an arc about two-thirds of the line
length from each end.
2. Locate and label the two points where the
arcs intersect.
3. Draw a line between the two labeled points.
This line now bisects the original line.

• Bisecting an Angle
1. Draw an arc at a convenient length from the
angle vertex.
2. Locate and label the two points where the
arc intersects the angle lines.
3. Draw the same size arc from each
intersection. At the intersection of the new
arcs, draw a line to the angle vertex.

• Drawing an Arc or Circle with Three Points
1. Draw a line from the middle point to each of
the other points.
2. Bisect each of the lines.
3. Where the bisect lines intersect is the center
of the circle.
4. Draw the circle using the new center point.

• Drawing Parallel Lines
1. Set the compass to the required parallel
distance.
2. Draw an arc from each end of the given line
on the same side of the line.
3. Connect the top of the arcs with a line. This
will be parallel to the given line.

• Drawing Perpendicular Lines
1. Draw an arc that crosses the given line in
two places from a given point on the line.
2. Draw a larger arc from each point where
the first arc intersects the given line.
3. Where the new arcs intersect, draw a
line to the original point on the line.

• Equally Dividing a Line
1. Draw a perpendicular line from one end of the
given line.
2. From the other end of the given line, measure a
distance that can be easily divided to the
perpendicular line.
3. Put tic marks on the angled line at the division
increments.
4. Transfer these marks perpendicularly to the given
line.

• Transferring Shapes
1. Label each point on the original shape.
2. Select a start point, move clockwise.
3. Set the compass to the distance from the start
point to the second point and lightly draw an arc at
the new location.
4. Proceed in the same manner to the other points on
the shape.
5. Connect all the points with a line once they have
been identified.

• Drawing a Square
1. Lightly draw center lines of a circle.
2. Lightly draw a circle with the diameter as the size of
the square.
3. Draw construction lines parallel to the center lines
and tangent to the circle.
4. If the four sides are equal, darken in the lines.
Square

• Drawing a Pentagon (five sides)
1. Lightly draw a circle with the diameter as the size
of the pentagon.
2. From the top of the circle, draw a 72° line from the
vertical axis.
3. Where this line intersects the circle, draw a
horizontal line. Set the compass for the length of
the line in the circle.
4. Transfer the compass distance around the circle
and connect the points.
Penta
gon

• Drawing a Hexagon (six sides)
of the hexagon.
2. Draw horizontal lines tangent to the top and
bottom of the circle.
3. Draw 60° lines from the horizontal lines, so they
are tangent to the circle.
4. Darken all lines. Hexag
on

• Drawing an Octagon (eight sides)
of the octagon.
2. Draw two horizontal and two vertical lines tangent
to the circle.
3. Draw 45° lines so they are tangent to the circle.
4. Darken all lines.
Octa
gon

• Arc within an Angle
1. Set the compass to the given radius.
2. Swing an arc from each line in the angle.
3. Draw tangent parallel lines from the given
angle lines to the arcs.
4. At the intersection of the lines, draw the
arc within the angle.

• Arc between a Curve and a Line
2. Swing an arc from the line and draw a
parallel line.
3. Swing an arc from the outside of the circle
and draw a concentric circle through the arc.
4. At the intersection of the line and the arc,
draw the new arc between the curve and the
line. Use the given radius.

• Arc between Two Curves
2. For each curve, swing an arc from the
outside of each curve and draw a concentric
arc through the first arc.
3. At the intersection of the arcs, draw a new
arc between the curves. Use the given
radius.

Conic Sections
• A conic section is formed by passing a plane
through a cone.
Triangle: formed when plane passes vertically
through the apex of a cone
Circle: formed when plane passes horizontally
through a cone
Ellipse: formed when plane passes through at an
incline

Conic Sections
• A conic section is formed by passing a plane
through a cone.
Parabola: formed when plane passes vertically
through a cone against its axis
Hyperbola: formed when plane passes vertically
through a cone

Conic Sections
• References to Drawing Irregular Shapes
(see Chapter 4)
– Drawing an ellipse
• Concentric circle method
• Trammel ellipse
• Foci ellipse
– Drawing a parabola
– Drawing a hyperbola

Conic Sections
• References to Drawing Irregular Shapes
(see Chapter 4)
– Drawing a spiral
– Drawing a helix
– Drawing an involute
– Drawing a cycloid curve

References
• http://www.engineering108.com/pages/Engineering
_graphics/Engineering_graphics_tutorials_free_dow
nload.html
• A text book of engineering graphics- Prof. P.J SHAH
• Engineering Drawing-N.D.Bhatt
• Engineering Drawing-P.S.Gill

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