Thesis Format
Monograph
Degree
Master of Science
Program
Applied Mathematics
Supervisor
Jeffrey, David
2nd Supervisor
Moreno Maza, Marc
Co-Supervisor
Abstract
Consider a complex analytic curve $X$ in $\mathbb{C}^2$, along with a specific point $p \in X$. The primary concern arises in approximating geometrically the curve $X$ precisely at the point $p$. Analogously, in introductory calculus, students learn to compute the tangent line to the graph of a function $y=f(x)$ at a given point $p=(x_*,f(x_*))$ by utilizing the derivative of $f$ at $x_*$.
For analytical convenience, we assume a local representation of the curve $X$ using a power series expansion. This representation centers the point $p$ at the origin $(0,0) \in \mathbb{C}^2$. Thus, our mathematical input becomes a Taylor series:
\[ f(x) = \sum_{i=1}^\infty c_ix^i \ . \]
Alternatively, one may conceptualize our input as a ``germ'' or a ``jet'' of sufficiently high order. This assumption of a local representation is grounded in the application of the implicit function theorem. The theorem, coupled with the specific nature of the questions posed, allows us to work with a localized description of $X$ that relies on only finitely many coefficients $\{c_i\}_{i=1}^\infty$.
An alternative perspective on defining a plane curve involves considering it as the set of points where an implicit polynomial function $F(x,y) \in \mathbb{C}[x,y]$ equals zero. This set of points is termed a ``variety,'' denoted as $\mathcal V(F(x,y)) = \{(x,y) \in \mathbb{C}^2 \mid F(x,y)=0\}$.
Given a degree $d$ and a sequence $\textbf{c}=(c_1,c_2,\ldots)$ defining the power series expansion, our objective is to ascertain the ``best approximation'' of the curve $\Gamma$ through a curve expressed as $\mathcal V(F(x,y))$, where $F(x,y)$ has a degree no greater than $d$. This specific curve is referred to as the ``degree $d$ osculating curve'' of the Taylor series, or equivalently, the ``degree $d$ osculating curve'' of $X$ at the point $p$.
Summary for Lay Audience
A curve is said to osculate a second curve if the two touch only at a point. A straight line tangent to a curve is a familiar example. The straight line osculates the curve at the point where it touches the curve. A second example will be familiar to some calculus students: an osculating circle. In this case, the circle not only touches a given curve, but also matches the curvature of the curve at the point of touching.
Osculating curves approximate the curve they are touching in the neighbourhood of the contact point. For this reason, they are used a lot in Computer Aided Design (CAD) to speed up calculations and to ensure that curves and surfaces remain smooth at places where they join.
The thesis develops a new way of calculating osculating curves, without being restricted to straight lines or circles. This allows formulae of greater generality than before to be computed.
Recommended Citation
Bahrami, Sepideh, "Osculating Curves" (2024). Electronic Thesis and Dissertation Repository. 10410.
https://ir.lib.uwo.ca/etd/10410
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