Instructor: Laura Gerold, PE               Catalog #10614113     Class # 22784, 24113, 24136, & 24138Class Start: January ...
   Project Proposals are due in two weeks on    February 22nd   Laura’s office hour will be cancelled next week    on Fe...
   For ten extra points, write a question for our    upcoming first exam (first exam is in one month on    March 7th)   ...
   The wrong scale is used for measurement   Architect Scale: 1/8 & ¼ scales are similar in    the middle and wrong numb...
There are ANSI/ASME standards for international and U.S. sheetsizes. Note that drawing sheet size is given as height width...
• Margins and Borders• Zones
   Units are not converted correctly   Not understanding the relationship between the    different sides of the scale  ...
   Engineer Scale       Actual Length        1. Measure actual length with 10 scale        2. 1:2 is half size. Use 20 s...
   Architect Scale       Actual Length        1. Measure actual length with 16 scale        2. 1:2 is half size. Divide ...
   Do the same exercise as 2.2 over again, using    the line lengths from 2.1.   Maximum of ten points will be rewarded...
THICK! Use your 7 mm mechanical pencil.
   How to Use the Ames Lettering Guide
   Very well! It is a course competency to be able    to “describe solids”   You need to be able to describe the followi...
   On a test, you may be asked to do one of the    following:       Draw one of the solids       Give the definition of...
   Three-Dimensional Geometry       Geometry Basics: 3D Geometry       Pyramids and Prisms       Cylinders and Cones
   Right Circular    Cylinder   This is the rotunda in    Birmingham, UK
   Sphere   Water Tower
   Right Pentagonal    Prism   The Pentagon
   Ellipsoid   Caravan Interior Light
   What object has a double curved surface and is    shaped like a donut?     A.   Ellipsoid     B.   Torus     C.   S...
   What object has two triangular bases and three    additional faces?     A.    Pyramid     B.    Torus     C.    Con...
   Make a list of solids that you saw today on    your way to class, in your house, or at work.   Sketch few of these sh...
• Triangles• Quadrilaterals• Polygons• Circles• Arcs
Three-dimensional figures are referredto as solids. Solids are bounded bythe surfaces that contain them. Thesesurfaces can...
   A two-dimensional figure, also called a plane or planar    figure, is a set of line segments or sides and curve    seg...
   A triangle is a plan figure bounded by three    straight lines   The sum of the interior angles is always 180    degr...
   A plane figure bounded by four straight sides   If opposite sides are parallel, the quadrilateral    is also a parall...
A Parallelogram is a four-sidedshape with two parallel sides.Parallelograms have the followingcharacteristics:• The opposi...
   A trapezium is defined by the properties it    does not have. It has no parallel sides. Any    quadrilateral drawn at ...
   A plane shape (two-    dimensional) with    straight sides.   Examples:    triangles, rectangles and    pentagon   N...
   Come up with a list of Polygons you see in    nature, at work, driving around, at home . . .    Etc.   Sketch a few o...
   A circle is a closed curve all points of which are the    same distance from a point called the center.   Circumferen...
   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 ...
   Concentric circles are circles that have their    centers at the same point   Eccentric circles are circles that do n...
   An arc is a portion of the circumference of a    circle   An arc could be a portion of some other curved    shape, su...
   Tangent is a line (or arc) which touches a circle    or ellipse at just one point. Below, the blue line    is a tangen...
   See Appendix for useful geometric formulas    (pages A-32 to A-37)
   From A ad B draw    equal arcs with radius    greater than half AB   Join Intersection D    and E with a straight    ...
   Draw line AB 2.3 inches long   Bisect this line using demonstrated method
Inclined lines can be drawn at standard angles with the 45° triangle and the30° x 60° triangle. The triangles are transpar...
For measuring or setting off angles other than those obtainable withtriangles, use a protractor. Plastic protractors are s...
   A Song about Angles
   1. Lightly draw arc CR   2. Lightly draw equal arcs r with radius    slightly larger than half BC, to intersect at D...
   1. Use any convenient radius R, and strike arcs    from centers A and A’   2. Strike equal arcs r, and draw side A’C’
   Draw any angle   Label its vertex C   Bisect the angle and transfer half the angle to    place its vertex at arbitra...
   AB is the line, CD is the given distance   Use CD distance as the radius and draw two    arcs with center points E an...
   Draw a line EF   Use distance FH equal to 1.2”   Draw a new line parallel to EF and distance    GH away             ...
     When the Point is Not on the Line (AB & P given)          From P, draw any convenient inclined line, PD on (a)     ...
     When the Point is on the Line (AB & P given)          With P as center and any radius, strike arcs to intersect AB ...
   Draw a line   Draw a point on the line   Draw a point through the point and    perpendicular to the line   Repeat p...
    Drawing a Triangle with Sides Given1.   Draw one side, C2.   Draw an arc with radius equal to A3.   Lightly draw an a...
    Drawing a right triangle with hypotenuse and     one side given1.   Given sides S and R2.   With AB as diameter equal...
   Draw a triangle with sides 3”, 3.35”, and 2.56.”   Bisect the three interior angles   The bisectors should meet at a...
   Math Made Easy: Measuring Angles (part 1)   Math Made Easy: Measure Angles (part 2)   Many angles can be laid out di...
    Tangent Method1.   Tangent = Opposite / Adjacent2.   Tangent of angle is y/x3.   Y = x tan4.   Assume value for x, ea...
    Sine Method1.    Sine = opposite / hypotenuse2.    Sine of angle is y/z3.    Draw line x to easy length, 104.    Find...
    Chord Method1.   Chord = Line with both endpoints on a circle2.   Draw line x to easy length, 103.   Draw an arc with...
   Draw two lines forming an angle of 35.5    degrees using the tangent, sine, and chord    methods   Draw two lines for...
   Side AB given   With A & B as centers and radius AB, lightly    construct arcs to intersect at C   Draw lines AC and...
   Draw a 2” line, AB   Construct an equilateral triangle
1.   One side AB, given2.   Draw a line perpendicular through point A3.   With A as center, AB as radius, draw an arc     ...
Diameters Method1. Given Circle2. Draw diameters at right angles to each other3. Intersections of diameters with circle ar...
   Lightly draw a 2.2” diameter circle   Inscribe a square inside the circle   Circumscribe a square around the circle
    Dividers Method      Divide the circumference of a circle into five equal       parts with dividers      Join point...
    Geometric Method     1.   Bisect radius OD at C     2.   Use C as the center and CA as the radius to lightly draw    ...
   Lightly draw a 5” diameter circle   Find the vertices of an inscribed regular    pentagon   Join vertices to form a ...
Each side of a hexagon is equal to the radius of the circumscribed circle          Use a compass                         C...
   Method 1 – Use a Compass     Each side of a hexagon is equal to the radius of the      circumscribed circle     Use ...
   Method 2 – Centerline Variation     Draw vertical and horizontal centerlines     With A & B as centers and radius eq...
   Lightly draw a 5” diameter circle   Inscribe a hexagon
   Given a circumscribed square, (the distance    “across flats”) draw the diagonals of the    square.   Use the corners...
   Lightly draw a 5” diameter circle   Inscribe an Octogon
   A,B, C are given points not on a straight line   Draw lines AB and BC (chords of the circle)   Draw perpendicular bi...
   Draw three points spaced apart randomly   Create a circle through the three points
   Method 1       This method uses the principle that any right triangle        inscribed in a circle cuts off a semicir...
   Method 2 – Reverse the procedure (longer)       Draw two nonparallel chords       Draw perpendicular bisectors.    ...
   Draw a circle with a random radius on its own    piece of paper   Give your circle to your neighbor   Find the cente...
   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...
Tangents
For small radii, such as 1/8R for fillets and rounds, it is not practicableto draw complete tangency constructions. Instea...
Connecting Two Parallel Lines   Connecting Two Nonparallel Lines
The conic sections are curves produced by planes intersecting a right circularcone.                Four types of curves ar...
If a circle is viewed with the line of sight perpendicular to the planeof the circle…     …the circle will appear as a cir...
The intersection of like-numbered lines will be points on the ellipse. Locate points in theremaining three quadrants in a ...
These ellipse guides are usually designated by the ellipse angle, theangle at which a circle is viewed to appear as an ell...
The curves are largely successive segments of geometric curves,such as the ellipse, parabola, hyperbola, and involute.
For many purposes, particularly where a small ellipse is required, use theapproximate circular arc method.
The curve of intersection between a right circular cone and a plane parallelto one of its elements is a parabola.
A helix is generated by a point moving around and along the surface of acylinder or cone with a uniform angular velocity a...
An involute is the path of a point on a string as the string unwinds from aline, polygon, or circle.
A cycloid is generated by a point P on the circumference of a circle thatrolls along a straight line                      ...
Like cycloids, these curves are used to form the outlines of certain gear teeth and are thereforeof practical importance i...
• Chapter 5 – Orthographic Projection• Project Proposal due in two weeks (February  22nd)
   On one of your sketches, answer the following    two questions:       What was the most useful thing that you learned...
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
Class 4 presentation posted
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Class 4 presentation posted

  1. 1. Instructor: Laura Gerold, PE Catalog #10614113 Class # 22784, 24113, 24136, & 24138Class Start: January 18, 2012 Class End: May 16, 2012
  2. 2.  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!
  3. 3.  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
  4. 4.  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)
  5. 5. There are ANSI/ASME standards for international and U.S. sheetsizes. Note that drawing sheet size is given as height width. Moststandard sheets use what is called a “landscape” orientation. * May also be used as a vertical sheet size at 11" tall by 8.5" wide.
  6. 6. • Margins and Borders• Zones
  7. 7.  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
  8. 8.  Engineer Scale  Actual Length 1. Measure actual length with 10 scale 2. 1:2 is half size. Use 20 scale and measure out length from step 1. Draw 3. 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 scale 2. 1:2 is half size. Use 40 scale to measure out length from step 1. Draw 3. 2:1 is double size. Use 10 scale to measure out length from step 1. Draw.
  9. 9.  Architect Scale  Actual Length 1. Measure actual length with 16 scale 2. 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 scale 2. 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.
  10. 10.  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)
  11. 11. THICK! Use your 7 mm mechanical pencil.
  12. 12.  How to Use the Ames Lettering Guide
  13. 13.  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
  14. 14.  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).
  15. 15.  Three-Dimensional Geometry  Geometry Basics: 3D Geometry  Pyramids and Prisms  Cylinders and Cones
  16. 16.  Right Circular Cylinder This is the rotunda in Birmingham, UK
  17. 17.  Sphere Water Tower
  18. 18.  Right Pentagonal Prism The Pentagon
  19. 19.  Ellipsoid Caravan Interior Light
  20. 20.  What object has a double curved surface and is shaped like a donut?  A. Ellipsoid  B. Torus  C. Sphere  D. Cylinder
  21. 21.  What object has two triangular bases and three additional faces?  A. Pyramid  B. Torus  C. Cone  D. Triangular Prism
  22. 22.  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
  23. 23. • Triangles• Quadrilaterals• Polygons• Circles• Arcs
  24. 24. 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 solidmay be, it is composed of combinationsof these basic types of surfaces.
  25. 25.  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.
  26. 26.  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
  27. 27.  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
  28. 28. A Parallelogram is a four-sidedshape with two parallel sides.Parallelograms have the followingcharacteristics:• The opposite sides are equal in length.• The opposite angles are equal.• The diagonals bisect each other.Examples are arectangle, rhombus, square.
  29. 29.  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.
  30. 30.  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)
  31. 31.  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
  32. 32.  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
  33. 33.  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.
  34. 34.  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
  35. 35.  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
  36. 36.  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.
  37. 37.  See Appendix for useful geometric formulas (pages A-32 to A-37)
  38. 38.  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
  39. 39.  Draw line AB 2.3 inches long Bisect this line using demonstrated method
  40. 40. Inclined lines can be drawn at standard angles with the 45° triangle and the30° x 60° triangle. The triangles are transparent so that you can see the linesof 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.
  41. 41. For measuring or setting off angles other than those obtainable withtriangles, use a protractor. Plastic protractors are satisfactory for most angular measurementsNickel silver protractors areavailable when high accuracyis required
  42. 42.  A Song about Angles
  43. 43.  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
  44. 44.  1. Use any convenient radius R, and strike arcs from centers A and A’ 2. Strike equal arcs r, and draw side A’C’
  45. 45.  Draw any angle Label its vertex C Bisect the angle and transfer half the angle to place its vertex at arbitrary point D
  46. 46.  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. T-square Method For Curves
  47. 47.  Draw a line EF Use distance FH equal to 1.2” Draw a new line parallel to EF and distance GH away T-square Method For Curves
  48. 48.  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
  49. 49.  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
  50. 50.  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
  51. 51.  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
  52. 52.  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
  53. 53.  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
  54. 54.  Math Made Easy: Measuring Angles (part 1) Math Made Easy: Measure Angles (part 2) Many angles can be laid out directly with the protractor.
  55. 55.  Tangent Method1. Tangent = Opposite / Adjacent2. Tangent of angle is y/x3. Y = x tan4. Assume value for x, easy such as 105. Look up tangent of and multiply by x (10)6. Measure y = 10 tan
  56. 56.  Sine Method1. Sine = opposite / hypotenuse2. Sine of angle is y/z3. Draw line x to easy length, 104. Find sine of angle , multiply by 105. Draw arc R = 10 sin
  57. 57.  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 ( /2)5. Draw length C
  58. 58.  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
  59. 59.  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
  60. 60.  Draw a 2” line, AB Construct an equilateral triangle
  61. 61. 1. 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 C4. With B and C as centers and AB as radius, lightly construct arcs to intersect at D5. Draw lines CD and BD
  62. 62. Diameters Method1. Given Circle2. Draw diameters at right angles to each other3. Intersections of diameters with circle are vertices of square4. Draw lines
  63. 63.  Lightly draw a 2.2” diameter circle Inscribe a square inside the circle Circumscribe a square around the circle
  64. 64.  Dividers Method  Divide the circumference of a circle into five equal parts with dividers  Join points with straight line Dividers Method Geometric Method
  65. 65.  Geometric Method 1. Bisect radius OD at C 2. Use C as the center and CA as the radius to lightly draw arc AE 3. With A as center and AE as radius draw arc EB 4. 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
  66. 66.  Lightly draw a 5” diameter circle Find the vertices of an inscribed regular pentagon Join vertices to form a five-pointed star
  67. 67. Each side of a hexagon is equal to the radius of the circumscribed circle Use a compass Centerline Variation Steps
  68. 68.  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
  69. 69.  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
  70. 70.  Lightly draw a 5” diameter circle Inscribe a hexagon
  71. 71.  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
  72. 72.  Lightly draw a 5” diameter circle Inscribe an Octogon
  73. 73.  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
  74. 74.  Draw three points spaced apart randomly Create a circle through the three points
  75. 75.  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
  76. 76.  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.
  77. 77.  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
  78. 78.  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
  79. 79. Tangents
  80. 80. For small radii, such as 1/8R for fillets and rounds, it is not practicableto draw complete tangency constructions. Instead, draw a 45° bisectorof the angle and locate the center of the arc by trial along this line
  81. 81. Connecting Two Parallel Lines Connecting Two Nonparallel Lines
  82. 82. The conic sections are curves produced by planes intersecting a right circularcone. Four types of curves are produced: the circle, ellipse, parabola, and hyperbola, according to the position of the planes.
  83. 83. 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
  84. 84. The intersection of like-numbered lines will be points on the ellipse. Locate points in theremaining 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.
  85. 85. These ellipse guides are usually designated by the ellipse angle, theangle at which a circle is viewed to appear as an ellipse.
  86. 86. The curves are largely successive segments of geometric curves,such as the ellipse, parabola, hyperbola, and involute.
  87. 87. For many purposes, particularly where a small ellipse is required, use theapproximate circular arc method.
  88. 88. The curve of intersection between a right circular cone and a plane parallelto one of its elements is a parabola.
  89. 89. A helix is generated by a point moving around and along the surface of acylinder or cone with a uniform angular velocity about the axis, and with auniform linear velocity about the axis, and with a uniform velocity in thedirection of the axis
  90. 90. An involute is the path of a point on a string as the string unwinds from aline, polygon, or circle.
  91. 91. A cycloid is generated by a point P on the circumference of a circle thatrolls along a straight line Cycloid
  92. 92. Like cycloids, these curves are used to form the outlines of certain gear teeth and are thereforeof practical importance in machine design.
  93. 93. • Chapter 5 – Orthographic Projection• Project Proposal due in two weeks (February 22nd)
  94. 94.  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?

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