2.2 Loci

• The meaning of a locus, and the rule that generates it
• Basic loci: circle, parallel lines, perpendicular bisector, angle bisector
• The ellipse: the two-focus property, the concentric circles method, the trammel method
• The cycloid, epicycloid and hypocycloid
• The involute of a circle and of a polygon
• The parabola: rectangle method and focus-directrix method
• The hyperbola
• The locus of a point on a link mechanism (2-link and 3-link)
• Application of loci in power mechanics

• define a locus as used in drawing
• describe types of locus used in drawing
• construct different types of locus used in technical drawing
• construct the locus of link mechanisms relevant to power mechanics
• appreciate the application of loci in power mechanics

“How is loci used in real life situation?”

“How are skills on loci applied in power mechanics?”

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1. What a locus is
   40 min · 1 practice tasks · quiz
   A locus as the path of a point obeying a rule, the plural loci, and the method that works for every locus in the course.
2. Constructing the basic loci
   40 min · 1 practice tasks · quiz
   Building the perpendicular bisector, the angle bisector and a parallel line with compasses, and why the construction is better than measuring.
3. The ellipse and its two foci
   40 min · 1 practice tasks · quiz
   The ellipse as a locus, the major and minor axes, the two foci, and the pins-and-string method that proves the rule.
4. The ellipse by concentric circles
   40 min · 1 practice tasks · quiz
   The standard construction from the two axis lengths: verticals from the outer circle, horizontals from the inner one.
5. The ellipse by the trammel method
   40 min · 1 practice tasks · quiz
   Marking a paper strip with the two half axes and sliding it across the axes to trace the ellipse, and when to choose it over the circles method.
6. What a cycloid is
   40 min · 1 practice tasks · quiz
   The path of a point on a rolling wheel, the base length, the arch height, and why the curve has a sharp cusp.
7. Constructing a cycloid
   40 min · 1 practice tasks · quiz
   The twelve-point construction: centre positions, horizontals from the circle divisions, and arcs of radius r.
8. Epicycloid and hypocycloid
   40 min · 1 practice tasks · quiz
   Rolling a circle on the outside and the inside of another circle, how the constructions change, and where the curves are used.
9. The involute of a circle
   40 min · 1 practice tasks · quiz
   The path of the end of a string unwound from a circle, the twelve-tangent construction, and why every gear tooth is this shape.
10. The involute of a polygon
    40 min · 1 practice tasks · quiz
    Unwinding a string from a square or triangle, why the curve is made of exact circular arcs, and how the radius grows at each corner.
11. The parabola by the rectangle method
    40 min · 1 practice tasks · quiz
    Building a parabola inside a rectangle from its span and rise, and where the shape turns up in real structures.
12. The parabola from focus and directrix
    40 min · 1 practice tasks · quiz
    The parabola as a locus of equal distances, the focus-directrix construction, and the reflector property that makes headlamps work.
13. The rectangular hyperbola
    40 min · 1 practice tasks · quiz
    The curve whose two distances always multiply to the same answer, how to build it from its asymptotes, and where it turns up in engine work.
14. The locus of a point on a two-link mechanism
    40 min · 1 practice tasks · quiz
    Plotting the path of a point on the connecting rod of a slider crank, and why an engine designer needs that path.
15. The locus of a point on a three-link mechanism
    40 min · 1 practice tasks · quiz
    Plotting a coupler point on a four-bar linkage, why the two arcs must always be crossed the same way, and how designers use coupler curves.
16. Loci at work in power mechanics
    40 min · 1 practice tasks · quiz
    Pulling the whole sub strand together: which curve turns up where, and the questions a locus answers before anything is made.