GOTHIC GEOMETRY
CHAPTER FOUR
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GEOMETRIC
CONSTRUCTION
1.
Identify the geometry that makes up basic 2D
drawings.
2.
Use board drafting or 2D CAD skills to create
technical figures.
3.
Describe the advantages of CAD contrasted
with drawing with manual instruments.
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Prof. Stehr: Design & Machine Elements
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OBJECTIVES
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GEOMETRY REVIEW

Triangles

Quadrilaterals

Polygons

Circles

Arcs
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BISECTING A LINE OR CIRCULAR ARC
Triangle and T-Square System
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Compass system
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BISECTING A LINE WITH TRIANGLE
AND T-SQUARE
From endpoints A and B, draw construction lines at 30°, 45°, or 60° with the
given line. Then, through their intersection, C, draw a line perpendicular to
the given line to locate the center D…
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Triangles
Inclined lines can be drawn at standard
angles with the 45° triangle and the 30° x
60° triangle.
The triangles are transparent so that you
can see the lines of the drawing through
them.
A useful combination of triangles is the
30° x 60° triangle with a long side of 10"
and a 45° triangle with each side 8" long.
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Any Angle in 15° Increments
With only a 30° x
60° triangle and a
45° triangle, you can
draw
any angle in 15°
increments
The bottom
of the triangle in
each case is resting
on the blade of the
T-square. Twenty-four
15° sectors are possible
with just these two
triangles used singly or in
combination.
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Protractors
For measuring or setting off angles other than those
obtainable with triangles, use a protractor.
Plastic protractors
are satisfactory for
most angular
measurements
Nickel silver protractors
are available when high
accuracy
is required
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ANGLES…
BISECTING
AN ANGLE
TRANSFERRING
AN ANGLE
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DRAWING A LINE PARALLEL TO A LINE
AND AT A GIVEN DISTANCE
T-square Method
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For Curves
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DRAWING A LINE THROUGH A POINT
AND PERPENDICULAR TO A LINE
When the Point Is Not on the Line
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When the Point Is on the Line
Prof. Stehr: Design & Machine Elements
T-square Method
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TRIANGLES…
DRAWING A
TRIANGLE WITH
SIDES GIVEN
DRAWING A RIGHT
TRIANGLE WITH
HYPOTENUSE AND
ONE SIDE GIVEN
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LAYING OUT AN ANGLE
 Tangent Method
 Sine Method
 Chord Method
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Many angles can be laid out directly with the triangle or protractor.
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DRAWING AN EQUILATERAL TRIANGLE
Alternative Method
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DRAWING A SQUARE



T-square Method
Diameters Method
Inscribed Circle Method
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You can use the AutoCAD Polygon
command to draw squares. The
Rectangle command is another quick
way to make a square in AutoCAD.
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DRAWING A REGULAR PENTAGON
Dividers Method
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Geometric Method
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DRAWING A HEXAGON
Each side of a hexagon is
equal to the radius of the
circumscribed circle
Use a compass
Centerline Variation
Steps
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Drawing an Octagon
Given an inscribed circle, or distance
“across flats”, use a T-square or
straightedge and a 45° triangle to draw
the eight sides tangent to the circle.
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Given a circumscribed square, (the distance
“across flats”) draw the diagonals of the
square. Then, use the corners of the square as
centers and half the diagonal as the radius to
draw arcs cutting the sides.
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FINDING THE CENTER OF A CIRCLE
Another method, slightly longer, is
to reverse the procedure.
This method uses the principle
that any right triangle inscribed
in a circle cuts off a semicircle.
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Draw two nonparallel chords and
draw perpendicular bisectors. The
intersection of the bisectors will be
the center of the circle.
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DRAWING TANGENTS TO TWO CIRCLES
AutoCAD software provides a
convenient object snap for
finding tangency.
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DRAWING AN ARC TANGENT TO A LINE OR ARC
AND THROUGH A POINT
Tangents
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DRAWING AN ARC TANGENT TO TWO
LINES AT RIGHT ANGLES
For small radii, such as 1/8R for fillets and rounds, it is
not practicable to draw complete tangency
constructions. Instead, draw a 45° bisector of the angle
and locate the center of the arc by trial along this line.
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DRAWING AN ARC TANGENT TO TWO LINES
AT ACUTE OR OBTUSE ANGLES
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DRAWING AN ARC TANGENT TO AN ARC
AND A STRAIGHT LINE
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DRAWING AN ARC TANGENT
TO TWO ARCS
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Drawing an Arc Tangent to Two Arcs and
Enclosing One or Both
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DRAWING AN OGEE CURVE
Connecting Two Parallel Lines
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Connecting Two Nonparallel Lines
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THE CONIC SECTIONS
The conic sections are curves produced by planes intersecting a right circular cone.
Four types of curves are produced: the circle, ellipse, parabola, and
hyperbola, according to the position of the planes.
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DRAWING A FOCI ELLIPSE
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DRAWING A CONCENTRIC CIRCLE ELLIPSE
If a circle is viewed with the line of sight perpendicular to the plane of the circle…
…the circle will appear as a circle, in true size and shape
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DRAWING A PARALLELOGRAM ELLIPSE
The intersection of like-numbered lines will be points on the ellipse. Locate points in the remaining three
quadrants in a similar manner. Sketch the ellipse lightly through the points, then darken the final ellipse with the
aid of an irregular curve.
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ELLIPSE TEMPLATES
These ellipse guides are usually designated by the ellipse angle, the angle
at which a circle is viewed to appear as an ellipse.
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Irregular Curves
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The curves are largely successive segments of
geometric curves, such as the ellipse, parabola,
hyperbola, and involute.
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DRAWING AN APPROXIMATE ELLIPSE
For many purposes, particularly where a small ellipse is required, use the
approximate circular arc method.
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DRAWING A PARABOLA
The curve of intersection between a right circular cone and a plane parallel
to one of its elements is a parabola.
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DRAWING A HELIX
A helix is generated by a point moving around and along the surface of a
cylinder or cone with a uniform angular velocity about the axis, and with a
uniform linear velocity about the axis, and with a uniform velocity in the
direction of the axis.
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DRAWING AN INVOLUTE
An involute is the path of a point on a string as the string unwinds from a
line, polygon, or circle.
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DRAWING A CYCLOID
A cycloid is generated by a point P on the circumference of a circle
that rolls along a straight line.
Cycloid
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DRAWING AN EPICYCLOID OR A
HYPOCYCLOID
Like cycloids, these curves are used to form the outlines of certain gear teeth
and are therefore of practical importance in machine design.
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End of
CHAPTER FOUR
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GEOMETRIC
CONSTRUCTION