1. Applied geometry
Flóra Hajdu
B406

2. Content Geometry definition Draw lines, arcs and circles Draw polygons
Draw an ellipse
Autocad basics
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3. Geometry The study of the size and shape of objects
The reationship of straight and curved lines in drawing shapes
Geometric constructions are made of individual lines and points drawn in proper relationship to one another
Accuracy!
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4. Geometry
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
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5. Draw parallel and perpendicular lines
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6. Draw a parallel line at a given distance from a line
Given line AB
Erect a perpendicular to AB
Space the given distance D from the line AB by scale measurement
Position a triangle so that one side of the triangle is parallel with the given line
Slide this triangle along the base to the point at the desired distance from the give line and draw the required line
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7. Bisect a line Given line AB
Set tha compass to a radius greater than ½ AB
Using centers at A and B draw intersecting arcs above and below line AB
A line drawn through the intersections will bisect AB and will be perpendicular to line AB
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
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8. Bisect an arc Given arc AB
Set tha compass to a radius greater than ½ AB
Using centers at A and B draw intersecting arcs above and below line AB
A line drawn through the intersections will divide AB into 2 equal parts
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
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9. Bisect an angle Given angle ABC
With center B and suitable radius draw an arc to intersect BC at D and BA at E
With centers D and E and equal radius draw arcs to intersect at F
Bisect angle ABC by drawing a line from point B through point F
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
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10. Divide a line into a given number of equal parts
Given line AB and the number of equal divisions desired N
Draw a line from A at an angle
Set the compasses on A and step the compasses along the line, marking off N arcs. Label the last one C
Draw line BC
With a triangle draw parallel lines to BC across the arcs
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11. Draw an arc tangent to the sides of an angle
Given radius R of the arc
Draw lines inside the angle parallel to the given lines at distance R
The centre of the arc will be at the section of the parallel lines (C)
Set the compass to radius R and with center C draw the arc tangent to the given sides
The tangent points A and B are found by drawing perpendiculars through C to the given lines
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
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12. Draw an arc tangent to a given circle and straight line
Given R, the radius of the arc and a circle with radius R1
Raw a line parallel to the given straight line between the circle and the line at distance R away from the given line
With the center of the circle as center and radius R+R1 draw an arc to cut the parallel straight line at C
With center C and radius R draw the required arc tangent to te circle and the straight line
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
Hajdu Flóra

13. Tangent lines from a point to a circle
Given point P and circle with radius R and center O
Draw line PO and bisect it (F point)
Draw a circle with radius FP (Thales’s theorem)
Mark the points where the circle with radius FP sections the circle with radius R T1 and T2
Draw lines T1P and T2P
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14. Internal tangents to 2 circles
Given circles with radius R1 and R2 (R1>R2) and center O1 and O2
Draw a circle with center O1 and radius R1-R2
Draw line O1O2 and bisect it (F point)
Draw a circle with radius FP (Thales’s theorem)
Mark the points where the circle with radius FP sections the circle with radius R1-R2
Connect the section points with O1 and expand the lines to getT11 and T12
Draw lines T11O1 and T12O2
Draw parallel lines to T11O1 and T12O2 through O2. Mark the section points T21 and T22
Draw lines T11T21 and T12T22
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15. External tangents to 2 circles
Given circles with radius R1 and R2 (R1>R2) and center O1 and O2
Draw a circle with center O1 and radius R1+R2
Draw line O1O2 and bisect it (F point)
Draw a circle with radius FP (Thales’s theorem)
Mark the points where the circle with radius FP sections the circle with radius R1-R2
Connect the section points with O1 and expand the lines to getT11 and T12
Draw lines T11O1 and T12O2
Draw parallel lines to T11O1 and T12O2 through O2. Mark the section points T21 and T22
Draw lines T11T22 and T12T21
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16. Draw an arc tangent to 2 circles
Given R, the radius of the arc and 2 circles with centers A and B and radius R1 and R2
With the center of circle A as center and radius R±R1 draw an arc in the area between the circles
With the center of circle B as center and radius R±R2 draw an arc in the area between to cut the other arc at C
With center C and radius R draw the required arc tangent to the given circles
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
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17. Draw an equilateral triangle
Draw one side of the triangle (AB)
With center A and radius AB draw an arc
With center C and radius AB draw an arc to cut the other arc at C
Draw straight lines AC and BC
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18. Draw a square Draw 2 perpendicular lines
Draw a quadrant with center A and radius R to get points D and B
Draw arcs with centers D and B and radius R to intersect at point C
Connect points A B C and D with straight lines
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19. Inscibe a pentagon in a given circle
Given point circle with center O and diameter AB
Bisect line OB at D
With center D and radius DC draw an arc to cut the diameter at E
With C as center and radius CE draw an arc to cut the circumference at F
Distance CF is one side of the pentagon
With radius CF as a chord mark off the remaining points on the circle
Connect the points with straight lines
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
Hajdu Flóra

20. Draw a hexagon inscibed in a circle (across corners)
Given center of circle and radius R
Mark a point anywhere on the circle. This will be the first vertex of the hexagon
Set the compass radius R. Make an arc across the circle. This will be the next vertex of the hexagon.
Move the compasses on to the next vertex and draw another arc.  Continue in this way until you have all six vertices.
Draw a line between each successive pairs of vertices
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21. Draw a hexagon circumsised a given circle (across flats)
Given center of circle O and radius R
Mark a point anywhere on the circle.
Set the compass radius R. Make an arc across the circle.
Move the compasses on to the next vertex and draw another arc.  Continue in this way until you have all six vertices. These will be the tangent points to the circle
Connect O with a straight line to the vertices
Draw the tangents to the circle
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22. Draw an ellipse – 2 circle method
Given major and minor diameter
Consrtuct 2 concentric circles with diameter equal to AB and CD
Divide the circles into a convenient number of equal parts
Where the radial lines intersect the outer circle draw parallel lines to CD inside the outer circle
Where the same radial lines intersects the inner circle draw a parallel to AB away from the inner circle
The intersection of these lines gives the points of the ellipse
Draw a smooth curve through these points
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
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23. Draw an ellipse – parallelogram method
Given major diameter AB and minor diameter CD
Construct a parallelogram
Divide CO into a number of equal parts
Divide CE into the same number of equal parts. Number the points from C
Draw a line from B to point 1 on line CE
Draw a line from A through point 1 on CO intersecting the previous line. The point if intersection will be one point on the ellipse
Proceed in the same manner to find other points on the ellipse
Draw a smooth curve through these points
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
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24. Draw an ellipse – parallelogram method (fast)
Given major diameter AB and minor diameter CD
Construct a parallelogram
Draw the diagonals of the ellipse
Draw a half circle with centre A and radius OC
Bisect the half circle
From the section point draw perpendicular lines to the edge of the parallelogram to get E and F points
From E and F draw parallel lines to AB to intersect the diagonals.
The section points are points of the ellipse
Draw a smooth curve through these points
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25. Draw an ellipse – four center method
Given major diameter CD and minor diameter AB
Join points A and C with a line
Draw an arc with point O as the center and radius OC and extend line OA to locate point E
Draw an arc with point A as the center and radius AE to locate point F
Draw the perpendicular bisector of line CF to locate points G and H
Mirror GH to the minor and minor axis to get point K and L
Draw arcs with G and K as centers and radius HA and EB to complete the ellipse
Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design
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26. Summary Plane geometry Lines Circles Polygons Ellipse Autocad basics
Next week: Pictorial representation, Axonometry: isometric, dimetric and oblique projection

27. Thank You for Your attention!