Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.
Last reviewed:October 2019
- The plane
- Loci and equations
- Equations of lines
- Angle between two lines
- Area of a triangle
- Linear combinations
- Conic sections
- Three-dimensional space
- Special surfaces
- Related Primary Literature
- Additional Reading
A branch of mathematics in which algebra is applied to the study of geometry. Because algebraic methods were first systematically applied to geometry in 1637 by the French philosopher-mathematician René Descartes, the subject is also called Cartesian geometry. The basis for an algebraic treatment of geometry is provided by the existence of a one-to-one correspondence between the elements, “points” of a directed line g, and the elements, “numbers,” that form the set of all real numbers. Such a correspondence establishes a coordinate system on g, and the number corresponding to a point of g is called its coordinate. The point O of g with coordinate zero is the origin of the coordinate system. A coordinate system on g is Cartesian provided that for each point P of g, its coordinate is the directed distance . Then all points of g on one side of O have positive coordinates (forming the positive half of g) and all points on the other side have negative coordinates. The point with coordinate 1 is called the unit point. Since the relation + = is clearly valid for each two points P, Q, of directed line g, then = − = q − p, where p and q are the coordinates of P and Q, respectively. Those points of g between P and Q, together with P, Q, form a line segment. In analytic geometry it is convenient to direct segments, writing PQ or QP accordingly as the segment is directed from P to Q or from Q to P, respectively. To find the coordinate of the point P that divides the segment P1 P2 in a given ratio r, put = r. Then (x − x1)/(x − x2) = r, where x1, x2, x are the coordinates of P1, P2, P, respectively, and solving for x gives x = (x1 − rx2)/(1 − r). Clearly r is negative for each point between P1, P2 and is positive for each point of g external to the segment. The midpoint of the segment divides it in the ratio −1, and hence its coordinate x = (x1 + x2)/2. See also: Mathematics
The content above is only an excerpt.
for your institution. Subscribe
To learn more about subscribing to AccessScience, or to request a no-risk trial of this award-winning scientific reference for your institution, fill in your information and a member of our Sales Team will contact you as soon as possible.
to your librarian. Recommend
Let your librarian know about the award-winning gateway to the most trustworthy and accurate scientific information.
AccessScience provides the most accurate and trustworthy scientific information available.
Recognized as an award-winning gateway to scientific knowledge, AccessScience is an amazing online resource that contains high-quality reference material written specifically for students. Contributors include more than 9000 highly qualified scientists and 43 Nobel Prize winners.
MORE THAN 8700 articles covering all major scientific disciplines and encompassing the McGraw-Hill Encyclopedia of Science & Technology and McGraw-Hill Yearbook of Science & Technology
115,000-PLUS definitions from the McGraw-Hill Dictionary of Scientific and Technical Terms
3000 biographies of notable scientific figures
MORE THAN 19,000 downloadable images and animations illustrating key topics
ENGAGING VIDEOS highlighting the life and work of award-winning scientists
SUGGESTIONS FOR FURTHER STUDY and additional readings to guide students to deeper understanding and research
LINKS TO CITABLE LITERATURE help students expand their knowledge using primary sources of information