Chapter 4: Geometry

4.19.2 Oblique spherical triangles
4.20 DIFFERENTIAL GEOMETRY
4.20.1 Curves
4.20.2 Surfaces
4.21 ANGLE CONVERSION
4.22 KNOTS UP TO EIGHT CROSSINGS
4.1 COORDINATE SYSTEMS IN THE PLANE
4.1.1 CONVENTION
When we talk about “the point with coordinates ´Ü ݵ” or “the curve with equation
Ý ´Üµ”, we always mean Cartesian coordinates. If a formula involves other
coordinates, this fact will be stated explicitly.
4.1.2 SUBSTITUTIONS AND TRANSFORMATIONS
Formulae for changes in coordinate systems can lead to confusion because (for example)
moving the coordinate axes up has the same effect on equations as moving
objects down while the axes stay x ed. (To read the next paragraph, you can move
your eyes down or slide the page up.)
To avoid confusion, we will carefully distinguish between transformations of
the plane and substitutions, as explained below. Similar considerations will apply to
transformations and substitutions in three dimensions (Section 4.8).
4.1.2.1 Substitutions
A substitution, orchange of coordinates, relates the coordinates of a point in one
coordinate system to those of the same point in a different coordinate system. Usually
one coordinate system has the superscript ¼ and the other does not, and we write
´
Ü Ü´Ü ¼ Ý ¼ µ
Ý Ý´Ü ¼ Ý ¼ or ´Ü ݵ ´Ü ¼ Ý ¼ µ (4.1.1)
µ
(where subscripts and primes are not derivatives, they are coordinates). This means:
given the equation of an object in the unprimed coordinate system, one obtains
the equation of the same object in the primed coordinate system by substituting
Ü´Ü ¼ Ý ¼ µ for Ü and Ý´Ü ¼ Ý ¼ µ for Ý in the equation. For instance, suppose the primed
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