Topic 1 of 7 · Form 5
Plane geometry: conics, loci and mechanisms
Curves formed by cutting a cone with a plane: circle, ellipse, parabola and hyperbola.
Notes for this topic 1 file, in order
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Conic sections
Curves formed by cutting a cone with a plane: circle, ellipse, parabola and hyperbola.
- Eccentricity e = distance to focus ÷ distance to directrix: ellipse e < 1, parabola e = 1, hyperbola e > 1.
- Ellipse constructions: concentric circles, rectangle method, trammel, and the foci method (sum of distances to foci = major axis).
- Parabola: rectangle method or focus–directrix method; used in reflectors, satellite dishes and bridge arches.
- Tangents and normals: the normal to an ellipse bisects the angle between the focal radii.
Loci of mechanisms
Plot points on linkages (four-bar chains, slider-crank mechanisms, trammels) through a complete cycle at equal intervals; join with a smooth curve.
Other curves
Involutes (gear teeth), cycloids, epicycloids and hypocycloids (rolling circles), Archimedean spirals (cams), and helices (threads and springs).
Cam profiles
Displacement diagrams for uniform velocity, simple harmonic motion and uniform acceleration; construct the cam profile for a knife-edge, roller or flat follower.
Check yourself
- What is the eccentricity of a parabola?
- Name one practical use of the involute curve.
- Which follower motion gives the smoothest cam action?
Show answers
- e = 1
- The profile of gear teeth.
- Simple harmonic motion (or uniform acceleration and retardation).
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