Geometrical construction

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• Geometrical construction

By Adzly Anuar, Zulkifli Bin Ahmad J.
Purbolaksono
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Contents

• Outcome of todays lecture
• Overview
• Basic geometrical constructions
• Bisect lines, angles, etc.
• Draw circles, hexagon, pentagon, etc.
• Draw arc tangents, etc.
• Conclusions

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Outcomes of todays lecture

• Able to explain
• Different types of basic geometrical construction
  techniques
• How to construct straight line, circle, hexagon,
  tangent and arc tangent

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Engineering Geometry

• Geometry provides the building blocks for the
  engineering design process.
• Engineering geometry is the basic geometric
  elements and forms used in engineering design.
• Coordinate system
• cartesian coordinate system
• polar coordinate system

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• Absolute coordinate Relative coordinate
• Right hand rule
• to determine positive direction of axis

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Geometry Elements

• Can be categorized as points, lines, surfaces,
  solids.
• Points, lines, circles and arcs are basic 2D
  geometric primitives.

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Geometry Elements

• Point - theoretical location that has neither
  width, height, nor depth. It describe an exact
  location in space. Represented as a small cross.
• Line - has length and direction, but not
  thickness. May be straight or curve or both.

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Circle

• Circle - is a single-curved-surface, all points
  of which are equidistant from one point, the
  center

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CIRCLE
Major components of a circle
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CIRCLE
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Tangent

• A line is tangent to a circle if it touches the
  circle at one and only one point.
• At exact point of tangency, a radius makes a
  right angle to the tangent line.
• Two curves are tangent to each other if they
  touch in one and only place.

Tangent point between a straight line a circle
Tangent point between two circles (curves)
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Basic Geometrical construction

• To develop the skill of
• Division of lines and angles
• Construction of tangents
• Blending of radii
• Accuracy is important, inaccuracy causes the
  constructions unusable

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Example How to draw?
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How to draw a straight line

• The most simple element
• Need to know
• the start point and end point or,
• Start point and length

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How to draw a circle

• First identify the centre point
• Draw the centre lines crossing at the centre
  point
• Set the compass to the required radius
• The sharp point is placed at the centre point and
  the circle is created using the pencil

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How to draw an arc

• Identify the centre point
• Identify the start and end points
• Use the compass similar to drawing a circle

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Bisect a straight line

• Bisecting a straight line
• To divide a line into two equal parts

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Drawing a perpendicular line from a point in a
line

• AB is the line, and C is the point on it
• With center C and any radius, describe equal arcs
  to cut AB at E and F
• From E and F describe equal arcs to intersect at
  D
• Join C and D to give the required perpendicular

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Bisecting an angle

• ABC is the given angle
• From B describe an arc to cut AB and BC at E and
  D respectively
• With centers E and D, draw equal arcs to
  intersect at F
• Join BF, the required bisector of the angle

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Drawing a line parallel to a given line at a
given distance from it

• AB is the given line, and c is the given distance
• From any two points well apart of AB, draw two
  arcs of radius equal to c
• Draw a line tangential to the two arcs to give
  the required line

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Constructing hexagon

• To construct a regular hexagon on a given line
• AB is the given line
• From A and B, and with radius AB, draw two equal
  arcs to intersect at O
• With radius OA or OB and center O draw a circle
• From A or B, using the same radius, step off arcs
  around the circle at C, D, E and F
• Join these points to complete the hexagon

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Constructing hexagon
Constructing a hexagon, given the distance
across flats
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Constructing pentagon
Constructing a pentagon, given the
diameter/radius of the circumscribe circle
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Constructing pentagon

• To construct a regular pentagon on a given line
• AB is the given line
• Bisect AB at C, erect a perpendicular at B, and
  mark off BD equal to AB
• With C as center and radius CD, describe an arc
  to intersect AB produced at E
• From A and B, and with radius AE, describe arcs
  to intersect at F
• With radius AB and centers A, B and F describe
  arcs to intersect at G and H
• Join FG, GA, FH and HB to complete the pentagon

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Draw tangent from a point to a circle

• draw straight line from centre point A of the
  circle to the given point B
• find the midpoint O of the line AB
• set the compass to the radius AO
• draw a circle or arc intersecting the circle A
• the crossing point is the tangent point

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Drawing a tangent to two given circles

• A and B are the centers of two given circles of
  radii r and R respectively
• With center B and radius R-r, describe a circle
• Bisect AB at X, and draw a semicircle on AB to
  cut circle R-r at C
• Join BC, and produce it to cut the larger circle
  at D
• Draw AE parallel to BD
• Join ED to give the required tangent

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Drawing an arc tangential to two straight lines

• AB and CB are the given lines, and c is the
  radius of the required arc
• Draw two lines parallel to the given lines at a
  distance c from them to intersect at D
• With centers D and radius c, draw an arc, which
  will be tangential to both given lines
• Erect perpendiculars at D to intersect AB and BC
  at E and F respectively. These are the points of
  tangency of the lines with the arc

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Drawing an arc tangential to two arcs (externally)

• A and B are the centers of the given arcs of
  radii a and b respectively c is the external arc
  radius
• From centers A and B, describe two arcs of radii
  a c and bc respectively to intersect at C
• With center C and radius c, describe an arc which
  will be tangential to the given arcs
• E and F are the points of tangency of the three
  arcs

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Drawing an arc tangential to two arcs (internally)

• A and B are the centers of the given arcs of
  radii a and b respectively c is the required
  tangential arc radius
• From centers A and B, describe two arcs of radii
  c-a and c-b respectively to intersect at C
• With center C and radius c, describe an arc which
  will be tangential to the given arcs
• E and F are the points of tangency of the three
  arcs

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Drawing an arc tangential to a line and another
arc

• A is the center of the given arc of radius a. BC
  is the given line, and b is the radius of the
  required arc
• From A, describe an arc with radius ab
• Draw a line parallel to BC and distant b, from it
  to intersect the arc ab at D
• From D, describe an arc of radius b, which will
  be tangential to the given line BC and the given
  arc a
• E and F are the points of tangency

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Drawing an arc tangential to two arcs and
enclosing one of them

• A and B as centers of two arcs of radius a and b
  respectively. Line c is the radius of the
  required arc
• With A and B as centers, describe arcs of radii
  ac and c-b respectively to intersect at C
• With center C and radius c, describe the required
  arc
• Join AC to intersect the curve at E, and produce
  CE to intersect the curve at F. Then E and F are
  the points of tangency of the three arcs

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Conclusions

• What has been covered today
• Engineering design
• Common geometrical elements
• How to draw?
• Examples of past drawings