GEOMETRIC CONSTRUCTION: CONIC SECTIONS

ENGINEERING DRAWING: LECTURE 4 - GEOMETRIC CONSTRUCTION: CONIC SECTIONS

1. Introduction to Conic Sections

Cone Formation: A cone is formed by rotating a right-angled triangle about its altitude (axis). The length/height of the cone equals the triangle's altitude, and the base radius equals the triangle's base. The apex angle of a cone is 2θ.

Conic Section Definition: A conic section is a section cut by a plane passing through a cone. The shape of the section depends on the plane's orientation relative to the cone and its axis. Possible shapes include a triangle, a circle, an ellipse, a parabola, or a hyperbola.

2. Types of Conic Sections by Plane Intersection

• Circle: Results when a plane passes through the cone, parallel with the base and perpendicular to the axis.
• Ellipse: A curve created when a plane passes through a right circular cone at an angle to the axis that is greater than the angle between the axis and the sides (elements).
• Parabola: The curve created when a plane intersects a right circular cone parallel to the side (elements) of the cone.
• Hyperbola: The curve of intersection created when a plane intersects a right circular cone and makes a smaller angle with the axis than do the sides (elements). This intersection produces two separate, unbounded curves.

3. Conic Sections as Loci of a Moving Point

• Circle: The set of points equidistant from a given point called the centre.
• Ellipse: The set of points in a plane whose sum of distances to two fixed points (foci) is a constant.
• Parabola: The set of points in a plane that are equidistant from a given point (focus) and a given line (directrix).
• Hyperbola: The set of points in a plane whose distances from two fixed points (foci) have a constant difference.

4. Ellipse Construction Methods

An ellipse is a curve traced by a point such that the sum of its distance from two fixed points (foci) is constant and equal to the major axis.

Given major axis (AB) and minor axis (CD), where F1 and F2 are foci:

• PF1 + PF2 = CF1 + CF2 = QF1 + QF2 = AB (Major axis)
• CF1 = CF2 = Half AB (Major axis)

Methods for drawing an ellipse:

1. Concentric Circle Method
2. Foci Method
3. Trammel Method
4. Oblong (Rectangle) Method

4.1. Ellipse - Concentric Circle Method

1. Construct two concentric circles with radii equal to ½ of the major and ½ of the minor axes.
2. Divide the circles into a convenient number of equal parts (e.g., 12).
3. Where radial lines intersect the inner circle, draw horizontal lines towards the larger circle.
4. Where radial lines cross the larger circle, draw vertical lines to meet the horizontal lines.
5. Draw a smooth curve through the intersections.

4.2. Ellipse - Foci Method

1. Set out the major (AB) and minor (CD) axes.
2. Find the focal points (F1 and F2) by drawing an arc from point C or D with radius OA (½ of major axis) to cut the major axis.
3. Select random points on the major axis between one focal point and the ellipse center (number them consecutively, spacing them closer at the ends).
4. Draw intersecting arcs whose sum of radii is AB from F1 and F2 (e.g., A1, B1 from F1 and F2; A2, B2 from F1 and F2, etc.).
5. Draw a smooth curve through the points of intersection of the curves.

4.3. Ellipse - Rectangle Method

1. Draw a rectangle with length and breadth equal to the major and minor axes.
2. Divide the two shorter sides of the rectangle into the same number of equal parts.
3. Divide the major axis into the same number of equal parts.
4. From the points where the minor axis crosses the edge of the rectangle, draw intersecting lines as shown.
5. Draw a neat curve through the intersections.

5. Parabola Construction Methods

5.1. Parabola - Rectangle Method

1. Draw a rectangle with dimensions equal to the rise (along the axis) and span (perpendicular to the axis).
2. Divide the span into two equal halves.
3. Divide half of the span into 'x' parts, such that the entire span is divided into 2x parts.
4. Divide the rise into the same number of 'x' parts.
5. Draw lines from the mid-span to the edges of the rectangle.
6. Draw vertical lines to intersect the corresponding lines.
7. Draw a smooth curve through the points of intersection.

5.2. Parabola - Offset Method

1. Given the enclosing rectangle (span AB, rise AC), construct a parallelogram.
2. Bisect AC to locate point O.
3. Divide OA into four equal parts (and similarly OB into four equal parts).
4. The offsets vary in length as the square of their distances from O. Since OA is divided into four equal parts, distance AC will be divided into 4² (16) equal divisions.
5. Locate points such that the point on the horizontal axis is the square of the value of the point on the vertical axis.
6. Draw a smooth curve through the points.

5.3. Parabola - Envelope Method

1. Draw lines AB and BC at right angles.
2. Divide the lines into the same number of equal parts.
3. Join corresponding points on each line.
4. The curve is drawn tangentially to the straight lines.

5.4. Parabola - Locus Method (Focus and Directrix)

1. Locate the focus (F) and directrix (D).
2. Locate the vertex (V) by bisecting the distance FD.
3. Draw lines parallel to the directrix.
4. With F as the center, and radii equal to the distance of the directrix from the parallel lines, draw arcs to cut the parallel lines.
5. Draw a smooth curve through the points of intersection.
6. D = Directrix, F = Focus, V = Vertex

6. Hyperbola Construction Methods

Eccentricity (e) is the ratio of the distances of a point to the focus and directrix:

• For hyperbola, e > 1
• For parabola, e = 1
• For ellipse, e < 1

6.1. Hyperbola: Locus Method 1 (Given Transverse Axis)

1. Draw a square whose sides equal the length of the transverse axis (AB).
2. Locate the center of the transverse axis.
3. From the center, with radii equal to half the length of the diagonal, swing arcs to intersect the horizontal line through the center of the square. This locates focus F1 and focus F2.
4. Progressing outward along the horizontal line, mark off equal spaces of arbitrary length from the focus points.
5. With A1 as radius, swing an arc from focus F1. With B1 as radius, swing an arc from focus F2.
6. Repeat the procedure for as many points as required (A2,B2; A3,B3; etc).
7. Using an irregular curve, carefully complete the hyperbola curve.

6.2. Hyperbola: Locus Method (Given Eccentricity)

To draw a hyperbola with eccentricity of say 3/2:

1. Locate the focus (F) and directrix (D).
2. Divide DF into five (3+2) parts and locate the vertex 2 divisions from D.
3. Draw lines parallel to the directrix.
4. With F as center, draw arcs to cut the parallel lines above and below the axis, taking the radius as: (Distance between directrix and line) * e (exDH in the diagram).

6.3. Hyperbola: Rectangular Method

Asymptotes: Lines which are tangents to the hyperbola at infinity.

Given asymptotes OA and OB and one point P on the curve:

1. Draw the asymptotes OA and OB and locate the given point, P.
2. Through point P, draw CD and EF parallel to the asymptotes.
3. Through any points on CD and EF, draw lines to locate other points on the curve.
4. Join the points with a smooth curve.

6.4. Hyperbola: Circumscribing Rectangular Method

The hyperbola has two branches, each with a focus and directrix, sharing the same eccentricity.

Given both vertices (V1 and V2) and a point P on one branch of the hyperbola:

1. Locate the vertices, V1 and V2, and the given point P.
2. Construct a rectangle through P and its associated vertex.
3. Divide the rectangle into two by the axis of symmetry.
4. Divide the two sides of the rectangles into the same number of equal parts.
5. Join the vertices to these points as shown to locate the points on one branch of the hyperbola.
6. Draw the second branch in a similar way, repeating the steps.