Curves in differential geometry

From Wikipedia, the free encyclopedia

This page covers mathematical example of curves in differential geometry.

Given a point p0 in R3 and a subinterval I of the real line,

\mathbf{\gamma}:t \mapsto \mathbf{p_0} = \begin{pmatrix} x_0\\ y_0\\ z_0\\ \end{pmatrix}\qquad (t \in I)

defines the constant curve, a parametric curve of class C∞. The image of the constant curve is the single point p. The curve is closed and analytic but not simple.

A slightly more complex example is the line. A parametric definition of a line through the points p0 and p1 (p0 ≠ p1 and p0,p1 ∈ R3) is given by

\mathbf{\gamma}:t \mapsto \mathbf{p_0} + t(\mathbf{p_1} - \mathbf{p_0})= \begin{pmatrix} x_0 + t (x_1 - x_0)\\ y_0 + t (y_1 - y_0) \\ z_0 + t (z_1 - z_0) \\ \end{pmatrix} \qquad (t \in I)

The image of the curve is a line. Note that

\mathbf{\gamma}:t \mapsto \mathbf{p_0} + t^3 (\mathbf{p_1} - \mathbf{p_0})= \begin{pmatrix} x_0 + t^3 (x_1 - x_0)\\ y_0 + t^3 (y_1 - y_0) \\ z_0 + t^3 (z_1 - z_0) \\ \end{pmatrix} \qquad (t \in I)

is a different curve but the image of both curves is the same line.

Given r, ω in R

\mathbf{\gamma}:t \mapsto \begin{pmatrix} r \cos (\omega t)\\ r \sin (\omega t)\\ t\\ \end{pmatrix}\qquad (t \in I)

defines a helix circling the z-axis.

Retrieved from "http://en.wikipedia.org../../../c/u/r/Curves_in_differential_geometry.html"
Advanced Search
Included Web Search Engines


Safe Search

close

Top Matching Results

Occasionally Search.com will highlight specialized results that are based on the context of your query. Examples of specialized results include specific links to news, images, or video.

Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links

This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.