Geometric Methods


To draw an approximate semi-ellipse by the five center method, Fig. 41, Plate 5. Having drawn the major axis E-H and the semi-minor axis C-D, complete the rectangle E-F-G-H-E. Draw E-C, and then F-B perpendicular to E-C and intersecting E-H at 0. Lay off D-S equal to D-C and with S-H as a diameter draw the semi-circle S-L-H. Measure D-T equal to K-L then with B as a center and a radius B-T describe an arc. With E and H as centers and a radius equal to A-D describe intersecting arcs at N and M. Through these points and center B draw lines B-Q and B-R. With O as a center and radius 0-E draw arc E-P. Then with N as a center and radius X-P draw arc P-Q. Then with B as a center and radius B-Q draw arc Q-C-R. Complete the ellipse similarly.

To draw a Tudor or pointed arch, Fig. 42, Plate 5, lay off the desired width or span A-B and the height or rise C-D. Select any desired radius for the small circle arc, say A-F and locate F and G. With F as a center and radius F-G swing an arc cutting the center line at E. Draw line F-E produced to meet a vertical line dropped from point G to locate H. With F as a center and radius F-H describe an arc; then with C as a center and a radius equal to A-F plus F-H, describe the small intersecting arc which locates point K. Draw line K-F produced through L. Now with fasa center and radius A-F draw arc A-L, then with A' as a center and radius K-L draw the arc L-C. Complete the arch similarly. This method produces the best arch when the radius of the smaller arc is equal to about one-fourth of the span but will work for other radii.

To draw a regular pentagon when given the distance from the center to a point, Fig. 43, Plate 5, draw the vertical and horizontal axes A-B and C-D intersecting at center E and draw the circumscribing circle. Then locate F, the middle point of C-E and with this as a center and a radius F-A, describe an arc cutting C-D at G. Then with A as a center and a radius A-G, swing the arc which locates point N on one side of the circle and point I on the other side. Then with H and I as centers and radius A-H, locate J and K. Connect A-H-K-J-I-A to form the pentagon.

To draw a regular hexagon proceed according to the method given on Plate 4 for dividing the circle into six equal parts. Connect the points on the circle using the 30-60 degree triangle.

To draw a regular octagon, Fig. 44, Plate 5, first draw the circumscribing circle with a diameter equal to the distance across points of the octagon. Draw the vertical and horizontal diameters and two others at 45 degrees with these. Connect the points where these lines cut the circumference of the circle.

To draw a circular intersection between two straight lines A-B and C-D, Fig. 45, Plate 5, draw E-F parallel to A-B and at a distance from A-B equal to the radius R of the connecting arc; then draw G-H similarly. Where these two lines intersect will be the center for the circle arc.