Project: My Favourite Curve

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 1. Plan (approx.): 


1. Name and description of a curve. Hystory. 

2. Equations of a curve (deduce/prove)

3. Other equations of a curve (parameterized, in polar coordinates, algebraic, etc.)
   and relations between them (prove)

4. Interesting points of a curve (intersecting with axes, max/min of x and y), asymptotes

5. Tangent line

6. Singular points, self-intersecting, angles for (semi-)tangent lines at this points 

7. Length (part or the whole)

8. Area (part or the whole)

9. Dependence on parameter (in equation)

10. Visualizing, Animation

11. Interesting properties of a curve

12. Constructions with a curve (parallel curve, inversion, etc.)

13. Applications of a curve

14. Curvature of a curve



see also: Fifty Famous Curves, Lots of Calculus Questions, And a Few Answers
http://facstaff.bloomu.edu/skokoska/curves.pdf
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 2. Themes of mini-project ''My favourite curve''


1.  Astroid, Deltoid 

2.  Cardioid, Limacon of Pascal, Nephroid

3.  Cartesian Ovals

4.  Cassinian Ovals, Lemniscate of Bernoulli

5.  Cissoid of Diocles, Strophoid

6.  Conchoid, Conchoid of de Sluze

7.  Epicycloid, Epitrochoid

8.  Hypocycloid, Hypotrochoid

9.  Algebraic spirals: Archimedean, hyperbolic,

10. Algebraic spirals: parabolic, logarythmic

11. Algebraic spirals: Fermat's spiral, Lituus

12. Lissajous Curves

13. Tractrix

14. Trisectrix of Maclaurin

15. Roses 

16. Descartes Ovals

17. Lame curves

18. Pearls of de Sluze or strophoid

19. Newton's Parabolas

20. Cycloid, trochoids

21. Folium of Descartes, Double Folium

22. Witch of Agnezi, Catenary

23. Tschirnhausen's Cubic, Cayley's sextic

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    Bibliography

1.  Gray A. Modern Differential Geometry of Curves and Surfaces with Mathematica. Springer-Verlag, 2-d ed. 1998.       
2.  do Carmo M. Differential Geometry of Curves and Surfaces. Birkhauser. 1992
3.  Rovenski V. Geometry of Curves and Surfaces with Maple, Birkhauser. 2000
4.  Cipolla R. & Giblin P. Visual Motion of Curves and Surfaces. 1999
5.  Yates R.C. Curves and their Properties. 1974
6.  Lockwood E.H. A Book of Curves. Cambridge University Press. 1991
7.  Lawrence J.D. A Catalog of Special Plane Curves. Dover Publications. 1972
8.  Zwikker C. The Advanced Geometry of Plane Curves and Their Applications. Dover. 2005  
9.  Dales J.B. Curves in Motion: Quilt Designs & Techniques. 1999 
10. Christopher G. Gibson, Chris Gibson Elementary Geometry of Differentiable Curves: 
    An Undergraduate Introduction. Cambridge. 2001 
11. Shikin E.V. Handbook and Atlas of Curves. CRC Press. 1995 
12. Savelov A.A., Ploskie krivye, Gos. izd. fiz.-mat. lit., Moskva, 1960 
13. Walker R. Algebraic curves, Princeton Univ. Press, Princeton, 1950 
14. Yates, R. CURVES AND THEIR PROPERTIES, The National Council of Teachers of Mathematics, 1952.
15. Gibson, C.G. Elementary Geometry of Algebraic Curves: An Undergraduate Introduction. Cambridge University Press, 1998.
16. Frost P. An Elementary Treatise on CURVE TRACING, 5th ed., Chelsea Publishing Company, 1960.
17. Rutter J.W. Geometry of Curves. CRC Press - 2000
18. Gibson C.G. Elementary Geometry of Differentiable Curves. Cambridge Univ. Press.
19. Nichols F. Curves. Child's Play-International - 1989
20. Arganbright D. Practical Handbook of Spreadsheet Curves and Geometric Constructions. 
    CRC Press, 1993, 197 P.  0622401 Class. QA483.A74 
21. Vasilyev, N. and V. Gutenmacher, Straight Lines and Curves, trans. Anjan Kundu, 1985

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Pursuit Curve, Bote lemniscate
Trident of Newton
Cochleoid
Equiangular Spiral
Kappa Curve
Serpentine
Sinusoidal Spirals