Geometric Constructions: A Comprehensive Guide
Geometric Constructions
Constructing Ovals and Ovoids
Oval Minor Axis
Given a line segment CD positioned vertically, bisect it to locate the center point O. Draw a line perpendicular to CD passing through O. Mark the points where this perpendicular intersects CD as N and M. Connect CM, CN, DM, and DN.
Oval Major Axis
Divide line segment AB into three equal parts. Label the division points as 1 and 2. Draw a circular arc centered at 1 and passing through A. Draw another circular arc centered at 2 and passing through B. Mark the points where these arcs intersect the line perpendicular to CD as N and M. Connect M and N with the corresponding points on AB.
Constructing an Oval with Two Axes
Given line segments AB and CD, bisect AB to find its midpoint O. Draw a vertical line from O, measuring the length of CD. From O, draw a circular arc with radius OA, intersecting the vertical line at point T. Connect C and T. Bisect AC and draw a perpendicular bisector. Mark the point where this bisector intersects the arc as C’. Connect C’ with O and D.
Ovoid Minor Axis
Given a line segment CD, draw a perpendicular bisector. From the center point O, draw a circular arc that intersects CD. Label the intersection points as P. Connect P with the endpoints of CD.
Ovoid Axis
Given a line segment AB, divide it into six equal parts. Draw a horizontal line from the second division point. From the midpoint of AB, draw a circular arc passing through A. Draw another circular arc centered at the second division point, passing through B, and mark the intersections with the horizontal line as R and S. Connect point 5 (the fifth division point on AB) with R and S.
Constructing an Ovoid with Two Axes
Given line segments AB and CD, position CD horizontally with its center at O. Draw a perpendicular bisector of CD. Mark the points where the bisector intersects the extension of CD as A and P. With P as the center, draw a circular arc passing through O. Mark the point where this arc intersects the lower part of the bisector as B. Using the radius PB, mark a point E on the extension of CD from D. Connect D with E and P. Extend the bisector to intersect the horizontal line at S. Mark a point S’ on the other side of the bisector, equidistant from O as S. Connect S and S’ with P.
Constructing Ellipses and Hyperbolas
Ellipse Points
Given line segments AB and CD, position CD horizontally and draw its perpendicular bisector. Using the length of AO, mark points on the extension of CD from C to locate the foci (F and F’). Divide the distance from F to O into eight equal parts and mark similar divisions on the other side. With B as the center and a radius equal to one division, draw an arc. Repeat this process for the other divisions and for the other focus F’, creating a series of arcs that intersect to form the ellipse.
Ellipse Affinity
Given line segments AB and CD, position CD horizontally and draw its perpendicular bisector. From the center point O, draw a circular arc passing through CD. Divide the arc into twelve equal parts. Connect O with each division point, creating a series of lines that intersect AB. Extend these lines and mark the points where they intersect the perpendicular bisector. Connect these intersection points with the corresponding points on AB to form the ellipse.
Projective Bundle Hyperbola
Given line segments AB and FF’, position them horizontally. From O, the midpoint of FF’, mark a point to the left and draw a perpendicular line equal in length to FF’ (both above and below the horizontal line). Repeat this process on the other side of O. From A and B, draw perpendicular lines intersecting the previously drawn lines. Label the intersection points as P. Divide each side of the resulting rectangles into two equal parts. Connect A with the division points on the top and bottom sides, and with the corresponding points on the other side. Repeat this process for B, creating the hyperbola.
Constructing Tangents
Tangent to an Ellipse from an External Point
Given an ellipse with major axis AB and foci F and F’, and an external point P, bisect FF’. Draw a circular arc centered at O (the midpoint of AB) with radius AO, intersecting the extension of CD at points P’. Connect P’ with F and F’. From F, draw circular arcs with radius AB. From P, draw an arc centered at F’. Connect F’ with the intersection points of the arcs, labeling them as 1 and 2. Draw the perpendicular bisectors of F’1 (labeled t1) and F’2 (labeled t2), both passing through P. Mark the points where t1 intersects the line connecting 1 and F’ as T1, and where t2 intersects the line connecting 2 and F’ as T2.
Tangent to a Hyperbola from an External Point
Given a hyperbola with transverse axis AB and conjugate axis CD, position CD vertically and draw its perpendicular bisector. From A and B, draw perpendicular lines with a height equal to CD. From the top corners of the resulting rectangle, draw circular arcs with F and F’ as centers. Using AB as the radius, draw a circular arc centered at F’. Draw a line perpendicular to the given line R, passing through F and intersecting the circular arc at points 1 and 2. Draw the perpendicular bisector of F1 (labeled t1). Draw a line parallel to R passing through point 1, intersecting t1 at T1. Connect 2 with F’ and draw the perpendicular bisector t2, intersecting the extension of F’2 at T2. Connect F with 2 (intersecting at T2) and F with 1 (intersecting at T1).
Tangent to a Parabola from an External Point
Given a parabola with focus F, directrix d, and an external point P, draw a vertical line (d) and a horizontal line representing the axis of the parabola. Measure the distance PF and draw an arc centered at P, intersecting the directrix at points 1 and 2. Connect these points with F. Draw the perpendicular bisector of F2, passing through P, and label the intersections with the axis as t1 and t2. Locate the points T1 and T2 where the lines connecting F with 1 and 2 intersect the horizontal axis.
Tangent to an Ellipse Parallel to a Given Line
Given an ellipse with major axis AB, foci F and F’, and a line R, draw circular arcs centered at F’ with radius AB. Using R as a reference, draw a line perpendicular to R passing through F and intersecting the circular arcs at points 1 and 2. Draw the perpendicular bisectors of F1 (labeled t1) and F2 (labeled t2). To find T1 and T2, connect F’ with 1 and 2, and mark the intersections with the respective bisectors.
Tangent to a Hyperbola Parallel to a Given Line
Given a hyperbola with transverse axis AB, conjugate axis CD, and a line R, position CD vertically and draw its perpendicular bisector. From A and B, draw perpendicular lines with a height equal to CD. From the top corners of the resulting rectangle, draw circular arcs with F and F’ as centers. Using AB as the radius, draw a circular arc centered at F’. Draw a line perpendicular to R passing through F and intersecting the circular arc at points 1 and 2. Draw the perpendicular bisector of F1 (labeled t1). Draw a line parallel to R passing through point 1, intersecting t1 at T1. Connect 2 with F’ and draw the perpendicular bisector t2, intersecting the extension of F’2 at T2. Connect F with 1 (intersecting at T1).