geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, curve sketching (or curve tracing) are techniques for producing a rough idea of overall shape of a

plane curve In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...

given its equation, without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features.

Basic techniques

The following are usually easy to carry out and give important clues as to the shape of a curve: *Determine the ''x'' and ''y'' intercepts of the curve. The ''x'' intercepts are found by setting ''y'' equal to 0 in the equation of the curve and solving for ''x''. Similarly, the ''y'' intercepts are found by setting ''x'' equal to 0 in the equation of the curve and solving for ''y''. *Determine the symmetry of the curve. If the exponent of ''x'' is always even in the equation of the curve then the ''y''-axis is an axis of

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

for the curve. Similarly, if the exponent of ''y'' is always even in the equation of the curve then the ''x''-axis is an axis of symmetry for the curve. If the sum of the degrees of ''x'' and ''y'' in each term is always even or always odd, then the curve is symmetric about the origin and the origin is called a ''center'' of the curve. *Determine any bounds on the values of ''x'' and ''y''. *If the curve passes through the origin then determine the tangent lines there. For algebraic curves, this can be done by removing all but the terms of lowest order from the equation and solving. *Similarly, removing all but the terms of highest order from the equation and solving gives the points where the curve meets the

line at infinity In geometry and topology, the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at ...

. *Determine the

asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

s of the curve. Also determine from which side the curve approaches the asymptotes and where the asymptotes intersect the curve. *Equate first and

second derivative In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...

s to 0 to find the

stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...

s and

inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph ...

s respectively. If the equation of the curve cannot be solved explicitly for ''x'' or ''y'', finding these derivatives requires

implicit differentiation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...

Newton's diagram

Newton's diagram (also known as ''Newton's parallelogram'', after

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...

) is a technique for determining the shape of an algebraic curve close to and far away from the origin. It consists of plotting (α, β) for each term ''Ax''^α''y''^β in the equation of the curve. The resulting diagram is then analyzed to produce information about the curve. Specifically, draw a diagonal line connecting two points on the diagram so that every other point is either on or to the right and above it. There is at least one such line if the curve passes through the origin. Let the equation of the line be ''q''α+''p''β=''r''. Suppose the curve is approximated by ''y''=''Cx^p/q'' near the origin. Then the term ''Ax''^α''y''^β is approximately ''Dx''^α+βp/q. The exponent is ''r/q'' when (α, β) is on the line and higher when it is above and to the right. Therefore, the significant terms near the origin under this assumption are only those lying on the line and the others may be ignored; it produces a simple approximate equation for the curve. There may be several such diagonal lines, each corresponding to one or more branches of the curve, and the approximate equations of the branches may be found by applying this method to each line in turn. For example, the folium of Descartes is defined by the equation :

x^3 + y^3 - 3 a x y = 0 \,

. Then Newton's diagram has points at (3, 0), (1, 1), and (0, 3). Two diagonal lines may be drawn as described above, 2α+β=3 and α+2β=3. These produce :

x^2 - 3 a y = 0 \,

y^2 - 3 a x = 0 \,

as approximate equations for the horizontal and vertical branches of the curve where they cross at the origin.

The analytical triangle

De Gua extended Newton's diagram to form a technique called the analytical triangle (or ''de Gua's triangle''). The points (α, β) are plotted as with Newton's diagram method but the line α+β=''n'', where ''n'' is the degree of the curve, is added to form a triangle which contains the diagram. This method considers all lines which bound the smallest convex polygon which contains the plotted points (see

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

Applications

* Streamline tracing in

fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...

Basic techniques

Newton's diagram

The analytical triangle

Applications

See also

Note

References

External links