Applied geometry Flra Hajdu B 406 hajdflsze hu

Applied geometry Flóra Hajdu B 406 hajdfl@sze. hu

Content • Geometry definition • Draw lines, arcs and circles • Draw polygons • Draw an ellipse 2021. 12. 18. Hajdu Flóra

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! 2021. 12. 18. Hajdu Flóra

Geometry Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design 2021. 12. 18. Hajdu Flóra

Draw parallel and perpendicular lines 2021. 12. 18. Hajdu Flóra

Draw a parallel line at a given distance from a line 1. Given line AB 2. Erect a perpendicular to AB 3. Space the given distance D from the line AB by scale measurement 4. Position a triangle so that one side of the triangle is parallel with the given line 5. Slide this triangle along the base to the point at the desired distance from the give line and draw the required line 2021. 12. 18. Hajdu Flóra

Bisect a line 1. Given line AB 2. Set tha compass to a radius greater than ½ AB 3. Using centers at A and B draw intersecting arcs above and below line AB 4. A line drawn through the intersections will bisect AB and will be perpendicular to line AB 2021. 12. 18. Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design Hajdu Flóra

Bisect an arc 1. Given arc AB 2. Set the compass to a radius greater than ½ AB 3. Using centers at A and B draw intersecting arcs above and below line AB 4. A line drawn through the intersections will divide AB into 2 equal parts 2021. 12. 18. Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design Hajdu Flóra

Bisect an angle 1. Given angle ABC 2. With center B and suitable radius draw an arc to intersect BC at D and BA at E 3. With centers D and E and equal radius draw arcs to intersect at F 4. Bisect angle ABC by drawing a line from point B through point F 2021. 12. 18. Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design Hajdu Flóra

Divide a line into a given number of equal parts 1. Given line AB and the number of equal divisions desired N 2. Draw a line from A at an angle 3. Set the compasses on A and step the compasses along the line, marking off N arcs. 4. Draw line B 1 5. With a triangle draw parallel lines to BC across the arcs 2021. 12. 18. Hajdu Flóra

Draw an arc tangent to the sides of an angle 1. 2. 3. 4. 5. 2021. 12. 18. 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 Hajdu Flóra Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design

Draw an arc tangent to a given circle and straight line 1. Given R, the radius of the arc and a circle with radius R 1 2. Draw a line parallel to the given straight line between the circle and the line at distance R away from the given line 3. With the center of the circle as center and radius R+R 1 draw an arc to cut the parallel straight line at C 4. With center C and radius R draw the required arc tangent to the circle and the straight line 2021. 12. 18. change Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design Hajdu Flóra

Tangent lines from a point to a circle 1. Given point P and circle with radius R and center O 2. Draw line PO and bisect it (F point) 3. Draw a circle with radius FP (Thales’s theorem) 4. Mark the points where the circle with radius FP sections the circle with radius R T 1 and T 2 5. Draw lines T 1 P and T 2 P 2021. 12. 18. Hajdu Flóra

External tangents to 2 circles 1. 2. 3. 4. 5. 6. 7. 8. 9. 2021. 12. 18. Given circles with radius R 1 and R 2 (R 1>R 2) and center O 1 and O 2 Draw a circle with center O 1 and radius R 1 -R 2 Draw line O 1 O 2 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 1 -R 2 Connect the section points with O 1 and expand the lines to get. T 11 and T 12 Draw lines T 11 O 1 and T 12 O 2 Draw parallel lines to T 11 O 1 and T 12 O 2 through O 2. Mark the section points T 21 and T 22 Draw lines T 11 T 21 and T 12 T 22 Hajdu Flóra

Internal tangents to 2 circles 1. 2. 3. 4. 5. 6. 7. 8. 9. 2021. 12. 18. Given circles with radius R 1 and R 2 (R 1>R 2) and center O 1 and O 2 Draw a circle with center O 1 and radius R 1+R 2 Draw line O 1 O 2 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 1 -R 2 Connect the section points with O 1 and expand the lines to get. T 11 and T 12 Draw lines T 11 O 1 and T 12 O 2 Draw parallel lines to T 11 O 1 and T 12 O 2 through O 2. Mark the section points T 21 and T 22 Draw lines T 11 T 22 and T 12 T 21 Hajdu Flóra

Draw an arc tangent to 2 circles 1. Given R, the radius of the arc and 2 circles with centers A and B and radius R 1 and R 2 2. With the center of circle A as center and radius R±R 1 draw an arc in the area between the circles 3. With the center of circle B as center and radius R±R 2 draw an arc in the area between to cut the other arc at C 4. With center C and radius R draw the required arc tangent to the given circles 2021. 12. 18. Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design Hajdu Flóra

Draw an equilateral triangle 1. Draw one side of the triangle (AB) 2. With center A and radius AB draw an arc 3. With center C and radius AB draw an arc to cut the other arc at C 4. Draw straight lines AC and BC 2021. 12. 18. Hajdu Flóra

Draw a square 1. Draw 2 perpendicular lines 2. Draw a quadrant with center A and radius a to get points D and B 3. Draw arcs with centers D and B and radius a to intersect at point C 4. Connect points A B C and D with straight lines 2021. 12. 18. Hajdu Flóra

Inscibe a pentagon in a given circle 1. 2. 3. 4. 5. 6. 7. 2021. 12. 18. 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

Draw a hexagon inscibed in a circle (across corners) 1. 2. 3. 4. 5. 2021. 12. 18. 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 Hajdu Flóra

Draw a hexagon circumsised a given circle (across flats) 1. 2. 3. 4. 5. 6. 2021. 12. 18. 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 Hajdu Flóra

Draw an ellipse – 2 circle method 1. 2. 3. 4. 5. 6. 7. 2021. 12. 18. 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 Hajdu Flóra Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design

Draw an ellipse – parallelogram method 1. 2. 3. 4. 5. 6. 7. 8. 2021. 12. 18. 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 of 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 Hajdu Flóra Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design

Draw an ellipse – parallelogram method (offset measurement) 1. 2. 3. 4. 5. 6. 7. 8. 9. 2021. 12. 18. 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 Hajdu Flóra

Draw an ellipse – four center method 1. 2. 3. 4. 5. 6. 7. 2021. 12. 18. 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 F Hajdu Flóra Source: C. Jensen, J. D. Helsel, D. R. Short: Engineering Drawing&Design

Summary • Plane geometry • Lines • Circles • Polygons • Ellipse • Autocad basics Next week: Pictorial representation: axonometric, oblique and perspective projection

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