Conic Sections and Drawing Methods

Conic Sections

• Ellipse: A conic section with an eccentricity less than 1.
• Parabola: A conic section with an eccentricity of 1.
• Hyperbola: A conic section with an eccentricity greater than 1.

Focus-Directrix Method

• Ellipse: Draw an ellipse with a focus a distance of 50 mm from the directrix and an eccentricity of 2/3;
• Parabola: Draw a parabola where the focus is 55 mm from the directrix;
• Hyperbola: Draw a hyperbola with an eccentricity of 4/3 where the focus is 60 mm from the directrix.

Concentric Method

• Draw an ellipse with a major axis of 60 mm and a minor axis of 40 mm

Oblong Method

• Draw an ellipse with conjugate axes of 60 mm and 40 mm, inclined at 75° to each other

Focus-Directrix or Eccentricity Method - Steps

• To draw an ellipse with focus 70 mm from directrix and eccentricity 3/4:
  • Divide the distance between the focus and the center into 7 equal parts (3 + 4)
  • The fourth point from the center is the point V and e = FV / CV = 3/4
  • Draw a line at 45° from the focus to meet CB produced at D
  • Construct perpendicular bisectors through the points V, 1, 2, 3... on the line and draw the ellipse.

Concentric Circle Method

• To draw an ellipse with major axis 70 mm and minor axis 40 mm:
  • Draw circles with the major and minor axes as diameters
  • Divide both circles into 12 equal parts and number them
  • Draw parallel lines from the points on the circles (1, 1', etc.) and draw a smooth curve connecting them.

Tangent and Normal

• Draw a tangent and normal for a point (P) on an ellipse:

  • Join PF and draw a line perpendicular to PF from the focus (F)
  • Join QP and that will be the tangent at P
  • The normal at P is perpendicular to QP.

• To obtain the tangent and normal to an ellipse when focus and directrix are unknown:

  • First, obtain the foci F and F" by cutting the arcs on the major axis.
  • Find the bisector of ∠FPF′; this is the normal.
  • Draw the tangent perpendicular to the normal at P.

Applications of Ellipse

• Arches
• Elliptical gears
• Bullet noses

Parabola

• A conic section with an eccentricity equal to 1
• Consists of a focus, directrix, and axis
• Any chord perpendicular to the axis of a parabola is a double ordinate
• The double ordinate passing through the focus is called the latus rectum
• The shortest distance from the vertex to any ordinate is the abscissa.

Methods for Generating a Parabola

• Focus-Directrix or Eccentricity Method: This method is applicable when the distance between the focus and directrix is known.
• Rectangle Method and Parallelogram Method: This method is applicable when the axis or base and the double ordinate of the parabola are known.
• Tangent Method: This method is applicable when the base and the inclination of tangents at the ends of the parabola are known.

Focus-Directrix or Eccentricity Method - Steps

• To draw a parabola with the focus 60 mm from the directrix:
  • Draw the directrix, axis, and mark the focus.
  • Mark the midpoint of the distance between the focus and the center as point V.
  • At V, erect a perpendicular VB = VF. Join CB.
  • Divide lines (RL and SL) into 6 equal parts.
  • Join the points 1-1', 2-2', 3-3', etc.
  • Draw a smooth curve tangent to 1-1', 2-2', 3-3', etc.

Tangent and Normal at a Point on a Parabola

• To draw a tangent and normal at point (P) for a parabola:
  • Draw the ordinate PS.
  • Mark T on LK such that TV = VS
  • Join TP and extend this to obtain the tangent
  • Draw the normal NM perpendicular to the tangent at P.

When Focus and Directrix are not Known

• To find the Tangent and Normal at a point, when focus and directrix are not known:
  • Draw the ordinate PQ and find the abscissa VQ.
  • Mark R on CA such that RV= VQ.
  • Draw the normal NM perpendicular to RP at P.

Finding the Focus and Directrix

• To find the focus and directrix of a parabola given its axis:
  • Mark any point P on the parabola and draw a perpendicular PQ to the axis.
  • Mark a point R on the axis such that RV = VQ.
  • The focus is the intersection of the perpendicular bisector of RP with the axis.
  • The directrix is perpendicular to the axis and passes through the point O where OV = VF.

Applications of Parabola

• Satellite dishes
• Reflectors
• Headlights
• Architectural design

Hyperbola

• A conic section with an eccentricity greater than 1.
• Has two asymptotes, focus, and directrix.

Summary

• Conic sections: Ellipses, parabolas and hyperbolas, are defined by their eccentricity.
• Focus-directrix method: A useful technique for constructing conic sections using the distance between the focus and the directrix.
• Focus-directrix method steps
  • For ellipse: Divide the distance between the focus and the center into 7 equal parts. The fourth point from the center is point V.
  • For parabola: Mark the midpoint between the focus and the center as point V.
  • For hyperbola: Similar to ellipse, divide the distance between the focus and the center to find point V.
• Concentric method: Used to construct an ellipse using two concentric circles.
• Tangent and Normal
  • Find the tangent and normal at a point on an ellipse by joining the point to the focus and drawing perpendiculars.
  • For a parabola, the tangent and normal can be found using the directrix and focus.
• Applications of Ellipses: Arches, elliptical gears, bullet noses.
• Parabola applications: Satellite dishes, headlights, reflectors.
• Hyperbola: A conic section with an eccentricity greater than 1, which has two asymptotes.