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2D ESSENTIALSInstructor: Laura Gerold, PE
Catalog #10614113Class # 22784, 24113, 24136, & 24138
Class Start: January 18, 2012 Class End: May 16, 2012
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QUESTIONS?
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Class Notes
Project Proposals are due in two weeks on February 22nd
Laura’s office hour will be cancelled next week on February 15th as she will be driving back from a 1 PM meeting in Milwaukee.
Please email any questions you may have about the homework or class!
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Extra Credit #1 For ten extra points, write a question for our
upcoming first exam (first exam is in one month on March 7th)
Question can be in any of the following formats Question with a drawing/sketch for an answer Essay Question Fill in the blank question True/False Multiple Choice
Question can cover any topics we covered in class so far. Also can include tonight and next week.
Please include your answer Question & Answer are due in two weeks on
February 22nd for extra ten points
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Keeping up with Technology
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Engineering Scale Common Errors
The wrong scale is used for measurement Architect Scale: 1/8 & ¼ scales are
similar in the middle and wrong numbers can be read off
Plans are printed out on a different size paper than was designed on and wrong scale used (check scales)
Plans don’t fit to scale as they were printed out wrong (check scales)
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STANDARD SHEETSThere are ANSI/ASME standards for international and U.S. sheet sizes. Note that drawing sheet size is given as height width. Most standard sheets use what is called a “landscape” orientation.
* May also be used as a vertical sheet size at 11" tall by 8.5" wide.
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Typical Sheet Sizes and Borders• Margins and Borders• Zones
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Engineering Scale Common Errors
Units are not converted correctly Not understanding the relationship
between the different sides of the scale Not understanding on engineer’s scale
that 1”=20’ (etc.) and on Architect’s scale that ¼” = 1’ (etc.)
Engineer scale is divided by tens, Architect scale is divided by twelve
Architect 16 Scale – Sixteen divisions per inch
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Exercise 2.2
Engineer Scale Actual Length
1. Measure actual length with 10 scale2. 1:2 is half size. Use 20 scale and measure out
length from step 1. Draw3. 2:1 is double size. Double measurement from
step 1, measure on 10 scale. Draw Using scale lengths to double
1. Measure length using 20 scale2. 1:2 is half size. Use 40 scale to measure out
length from step 1. Draw3. 2:1 is double size. Use 10 scale to measure out
length from step 1. Draw.
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Exercise 2.2
Architect Scale Actual Length
1. Measure actual length with 16 scale2. 1:2 is half size. Divide length from Step 1 in 2.
Use 16 scale and measure out length. Draw.3. 2:1 is double size. Double measurement from
step 1, measure on 16 scale. Draw. Using scale lengths to double
1. Measure length using 1/4 scale2. 1:2 is half size. Use 1/8 scale to measure out
length from step 1. Draw.3. 2:1 is double size. Use ½ scale to measure out
length from step 1. Draw.
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Extra Credit #2
Do the same exercise as 2.2 over again, using the line lengths from 2.1.
Maximum of ten points will be rewarded Some tips
Label lines 1, 2, 3, etc. Explain which scale you used and label on lines
(engineer or architect, 20, 1/2 ) Include your measurements of the original lines
Highly recommended for students who were deducted points on this exercise (and bonus for those scale experts who were not)
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What Line Weight Should be used for Title Blocks and Borders?
THICK! Use your 7 mm mechanical pencil.
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Ames Lettering Guide
How to Use the Ames Lettering Guide
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How well do we need to know solids?
Very well! It is a course competency to be able to “describe solids”
You need to be able to describe the following: Prisms Cylinders Pyramids Cones Spheres Torus Ellipsoids
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How well do we need to know solids?
On a test, you may be asked to do one of the following: Draw one of the solids Give the definition of one of the solids (essay,
fill in the blank, true/false, or multiple choice (like questions 2 & 3 on your technical sketching worksheet homework).
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Solids
Three-Dimensional Geometry Geometry Basics: 3D Geometry Pyramids and Prisms Cylinders and Cones
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Pop Quiz: What Solid?
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Pop Quiz: What Solid?
Right Circular Cylinder
This is the rotunda in Birmingham, UK
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Pop Quiz: What Solid?
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Pop Quiz: What Solid?
Sphere Water Tower
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Pop Quiz: What Solid?
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Pop Quiz: What Solid?
Right Pentagonal Prism
The Pentagon
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Pop Quiz: What Solid?
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Pop Quiz: What Solid?
Ellipsoid Caravan Interior
Light
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Pop Quiz: What Solid?
What object has a double curved surface and is shaped like a donut? A. Ellipsoid B. Torus C. Sphere D. Cylinder
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Pop Quiz: What Solid?
What object has two triangular bases and three additional faces? A. Pyramid B. Torus C. Cone D. Triangular Prism
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Solids Group Activity
Make a list of solids that you saw today on your way to class, in your house, or at work.
Sketch few of these shapes Present
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CHAPTER 4 – GEOMETRIC
CONSTRUCTION
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GEOMETRY REVIEW
• Triangles
• Quadrilaterals
• Polygons
• Circles
• Arcs
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UNDERSTANDING SOLID OBJECTS
Three-dimensional figures are referredto as solids. Solids are bounded bythe surfaces that contain them. Thesesurfaces can be one of the following fourtypes:
• Planar (flat)• Single curved (one curved surface)• Double curved (two curved surfaces)• Warped (uneven surface)Regardless of how complex a solid may be, it is composed of combinations of these basic types of surfaces.
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What are Plane Figures?
A two-dimensional figure, also called a plane or planar figure, is a set of line segments or sides and curve segments or arcs, all lying in a single plane. The sides and arcs are called the edges of the figure. The edges are one-dimensional, but they lie in the plane, which is two-dimensional.
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Triangles
A triangle is a plan figure bounded by three straight lines
The sum of the interior angles is always 180 degrees
A right triangle has one 90 degree angle
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Quadrilaterals
A plane figure bounded by four straight sides
If opposite sides are parallel, the quadrilateral is also a parallelogram
A Trapezoid is a quadrilateral which has at least one pair of parallel sides
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Parallelograms
A Parallelogram is a four-sided shape with two parallel sides.
Parallelograms have the following characteristics:• The opposite sides are equal
in length.• The opposite angles are
equal.• The diagonals bisect each
other.
Examples are a rectangle, rhombus, square.
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Trapezium
A trapezium is defined by the properties it does not have. It has no parallel sides. Any quadrilateral drawn at random would probably be a trapezium.
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Polygon
A plane shape (two-dimensional) with straight sides.
Examples: triangles, rectangles and pentagon
Note: a circle is not a polygon because it has a curved side
Be prepared to define a polygon up to an eight-sided figure (Figure 4.3, page 125)
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Group Activity
Come up with a list of Polygons you see in nature, at work, driving around, at home . . . Etc.
Sketch a few of these shapes. Present shapes
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Circles A circle is a closed curve all points of which
are the same distance from a point called the center.
Circumference refers to the distance around the circle (equal to pi (3.1416 multiplied by the diameter)
A diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle
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Circles A radius of a circle is the line from the
center of a circle to a point on the circle. The quadrant of a circle is a quarter of a
circle (made by two radiuses at right angles and the connecting arc)
A chord of a circle is a line that links two points on a circle or curve.
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Circles
Concentric circles are circles that have their centers at the same point
Eccentric circles are circles that do not have their centers at the same point
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Arcs
An arc is a portion of the circumference of a circle
An arc could be a portion of some other curved shape, such as an ellipse, but it almost always refers to a circle
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Tangent
Tangent is a line (or arc) which touches a circle or ellipse at just one point. Below, the blue line is a tangent to the circle c. Note the radius to the point of tangency is always perpendicular to the tangent line.
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Geometric Formulas
See Appendix for useful geometric formulas (pages A-32 to A-37)
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BISECTING A LINE OR CIRCULAR ARC
From A ad B draw equal arcs with radius greater than half AB
Join Intersection D and E with a straight line to locate center C
Compass system
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BISECTING A LINE OR CIRCULAR ARC
Draw line AB 2.3 inches long Bisect this line using demonstrated
method
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TrianglesInclined 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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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 accuracyis required
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BISECTING AN ANGLE
1. Lightly draw arc CR 2. Lightly draw equal arcs r with radius
slightly larger than half BC, to intersect at D
3. Draw line AD, which bisects the angle
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TRANSFERRING AN ANGLE
1. Use any convenient radius R, and strike arcs from centers A and A’
2. Strike equal arcs r, and draw side A’C’
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Angles
Draw any angle Label its vertex C Bisect the angle and transfer half the
angle to place its vertex at arbitrary point D
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DRAWING A LINE PARALLEL TO A LINE AND AT A GIVEN DISTANCE AB is the line, CD is the given distance Use CD distance as the radius and draw
two arcs with center points E and F near the ends of the line AB
Line GH (tangent to the arcs) is the required line.
For CurvesT-square Method
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DRAWING A LINE PARALLEL TO A LINE AND AT A GIVEN DISTANCE Draw a line EF Use distance FH equal to 1.2” Draw a new line parallel to EF and
distance GH away
For CurvesT-square Method
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DRAWING A LINE THROUGH A POINT AND PERPENDICULAR TO A LINE
When the Point is Not on the Line (AB & P given) From P, draw any convenient inclined line, PD on (a) Find center, C, of line PD Draw arc with radius CP Line EP is required perpendicular P as center, draw an arc to intersect AB at C and D (b) With C & D as centers and radius slightly greater than
half CD, draw arcs to intersect at E Line PE is required perpendicular
When the Point Is Not on the Line When the Point Is on the Line T-square Method
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DRAWING A LINE THROUGH A POINT AND PERPENDICULAR TO A LINE
When the Point is on the Line (AB & P given) With P as center and any radius, strike arcs to intersect
AB at D and G (c) With D and G as centers and radius slightly greater than
half DG, draw equal arcs to intersect at F. Line PF is the required perpendicular
When the Point Is Not on the Line When the Point Is on the Line T-square Method
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DRAWING A LINE THROUGH A POINT AND PERPENDICULAR TO A LINE
Draw a line Draw a point on the line Draw a point through the point and
perpendicular to the line Repeat process, but this time put the
point not on the line
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TRIANGLES
Drawing a Triangle with Sides Given1. Draw one side, C2. Draw an arc with radius equal to A3. Lightly draw an arc with radius equal to
B4. Draw sides A and B from the intersection
of the arcs
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TRIANGLES Drawing a right triangle with hypotenuse
and one side given1. Given sides S and R2. With AB as diameter equal to S, draw a
semicircle3. With A as center, R as radius, draw an arc
intersecting the semicircle C.4. Draw AC and CB
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TRIANGLES
Draw a triangle with sides 3”, 3.35”, and 2.56.”
Bisect the three interior angles The bisectors should meet at a point Draw a circle inscribed in the triangle
with the point where the bisectors meet in the center
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Measuring Angles with a Protractor
Math Made Easy: Measuring Angles (part 1)
Math Made Easy: Measure Angles (part 2)
Many angles can be laid out directly with the protractor.
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LAYING OUT AN ANGLE Tangent Method1. Tangent = Opposite / Adjacent2. Tangent of angle q is y/x3. Y = x tan q4. Assume value for x, easy such as 105. Look up tangent of q and multiply by x
(10)6. Measure y = 10 tan q
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LAYING OUT AN ANGLE Sine Method1. Sine = opposite / hypotenuse2. Sine of angle q is y/z3. Draw line x to easy length, 104. Find sine of angle q, multiply by 105. Draw arc R = 10 sin q
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LAYING OUT AN ANGLE Chord Method1. Chord = Line with both endpoints on a
circle2. Draw line x to easy length, 103. Draw an arc with convenient radius R4. C = 2 sin (q/2)5. Draw length C
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LAYING OUT AN ANGLE
Draw two lines forming an angle of 35.5 degrees using the tangent, sine, and chord methods
Draw two lines forming an angle of 40 degrees using your protractor
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DRAWING AN EQUILATERALTRIANGLE
Side AB given With A & B as centers and radius AB,
lightly construct arcs to intersect at C Draw lines AC and BC to complete
triangle
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DRAWING AN EQUILATERALTRIANGLE
Draw a 2” line, AB Construct an equilateral triangle
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DRAWING A SQUARE1. One side AB, given2. Draw a line perpendicular through point
A3. With A as center, AB as radius, draw an
arc intersecting the perpendicular line at C
4. With B and C as centers and AB as radius, lightly construct arcs to intersect at D
5. Draw lines CD and BD
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DRAWING A SQUAREDiameters Method1. Given Circle2. Draw diameters at right angles to each
other3. Intersections of diameters with circle are
vertices of square4. Draw lines
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DRAWING A SQUARE
Lightly draw a 2.2” diameter circle Inscribe a square inside the circle Circumscribe a square around the circle
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DRAWING A REGULAR PENTAGON Dividers Method
Divide the circumference of a circle into five equal parts with dividers
Join points with straight line
Dividers Method Geometric Method
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DRAWING A REGULAR PENTAGON Geometric Method
1. Bisect radius OD at C2. Use C as the center and CA as the radius to lightly
draw arc AE3. With A as center and AE as radius draw arc EB4. Draw line AB, then measure off distances AB
around the circumference of the circle, and draw the sides of the Pentagon through these points
Dividers Method Geometric Method
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DRAWING A REGULAR PENTAGON Lightly draw a 5” diameter circle Find the vertices of an inscribed regular
pentagon Join vertices to form a five-pointed star
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DRAWING A HEXAGONEach side of a hexagon is equal to the radius of the circumscribed circle
Use a compass Centerline Variation
Steps
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DRAWING A HEXAGON Method 1 – Use a Compass
Each side of a hexagon is equal to the radius of the circumscribed circle
Use the radius of the circle to mark the six sides of the hexagon around the circle
Connect the points with straight lines Check that the opposite sides are parallel
Use a compass
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DRAWING A HEXAGON Method 2 – Centerline Variation
Draw vertical and horizontal centerlines With A & B as centers and radius equal to that
of the circle, draw arcs to intersect the circle at C, D, E, and F
Complete the hexagon
Centerline Variation
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DRAWING A HEXAGON
Lightly draw a 5” diameter circle Inscribe a hexagon
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Drawing an Octagon Given a circumscribed square, (the
distance “across flats”) draw the diagonals of the square.
Use the corners of the square as centers and half the diagonal as the radius to draw arcs cutting the sides
Use a straight edge to draw the eight sides
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Drawing an Octagon
Lightly draw a 5” diameter circle Inscribe an Octogon
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DRAWING A CIRCLE THROUGH 3 POINTS
A,B, C are given points not on a straight line
Draw lines AB and BC (chords of the circle)
Draw perpendicular bisectors EO and DO intersecting at O
With center at ), draw circle through the points
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DRAWING A CIRCLE THROUGH 3 POINTS
Draw three points spaced apart randomly Create a circle through the three points
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FINDING THE CENTER OF A CIRCLE
Method 1 This method uses the principle that any right
triangle inscribed in a circle cuts off a semicircle
Draw any cord AB, preferably horizontal Draw perpendiculars from A and B, cutting the
circle at D and E Draw diagonals DB and EA whose intersection
C will be the center of the circle
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FINDING THE CENTER OF A CIRCLE
Method 2 – Reverse the procedure (longer) Draw two nonparallel chords Draw perpendicular bisectors. The intersection of the bisectors will be the
center of the circle.
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FINDING THE CENTER OF A CIRCLE
Draw a circle with a random radius on its own piece of paper
Give your circle to your neighbor Find the center of the circle given to you
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DRAWING A CIRCLE TANGENT TO A LINE AT A GIVEN POINT
Given a line AB and a point P on the line At P, draw a perpendicular to the line Mark the radius of the required circle on
the perpendicular Draw a circle with radius CP
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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 OROBTUSE 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 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 planeof 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 CurvesThe 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 PARABOLAThe 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 HELIXA 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 INVOLUTEAn 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 thereforeof practical importance in machine design.
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What’s Next?• Chapter 5 – Orthographic Projection• Project Proposal due in two weeks
(February 22nd)
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Questions?
On one of your sketches, answer the following two questions: What was the most useful thing that you
learned today? What do you still have questions about?