Book contents
- Frontmatter
- Contents
- Dedication
- PREFACE
- HISTORICAL INTRODUCTION
- NOTATION
- PART I SPECIAL CURVES
- PART II WAYS OF FINDING NEW CURVES
- 14 Conchoids
- 15 Cissoids
- 16 Strophoids
- 17 Roulettes
- 18 Pedal Curves
- 19 Negative Pedals
- 20 Glissettes
- 21 Evolutes and Involutes
- 22 Spirals
- 23 Inversion
- 24 Caustic Curves
- 25 Bipolar Coordinates
- FURTHER READING
- GLOSSARY
- INDEX OF NAMES
- INDEX OF SUBJECTS
- Frontmatter
- Contents
- Dedication
- PREFACE
- HISTORICAL INTRODUCTION
- NOTATION
- PART I SPECIAL CURVES
- PART II WAYS OF FINDING NEW CURVES
- 14 Conchoids
- 15 Cissoids
- 16 Strophoids
- 17 Roulettes
- 18 Pedal Curves
- 19 Negative Pedals
- 20 Glissettes
- 21 Evolutes and Involutes
- 22 Spirals
- 23 Inversion
- 24 Caustic Curves
- 25 Bipolar Coordinates
- FURTHER READING
- GLOSSARY
- INDEX OF NAMES
- INDEX OF SUBJECTS
Summary
Definition
Let S and S' be any two curves and let A be a. fixed point. A straight line is drawn through A cutting S and S' at Q and R respectively, and a point P is found in the line such that AP = QR, these lengths being measured in the direction indicated by the order of the letters. Then the locus of P is called the cissoid of S and S' with respect to A.
Thus the cissoid of two concentric circles, radii r1r2, with respect to their common centre is a circle with the same centre and radius |r1 –r2|.
The Cissoid of Diodes
This is the cissoid of a circle and a straight line touching it, with respect to the point on the circumference of the circle diametrically opposite to the point of contact. In Fig. 89, A is the fixed point, S and S' are the circle and the tangent at 5, and AP = QR.
This curve may be used for finding two mean proportionals between two given lengths. In Fig. 89, OU is the first of two mean proportionals between OC and OL; or, if the circle is of unit radius, the measure of OU is the cube root of that of OL. (Hint for proof : Let AO be a and let angle OAP be θ. Express coordinates of U,P and L in terms of a and θ.)
For other properties of this curve, see the summary given below.
- Type
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- Information
- Book of Curves , pp. 131 - 134Publisher: Cambridge University PressPrint publication year: 1961