algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, the conchoids of de Sluze are a

family Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...

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 ...

s studied in 1662 by Walloon mathematician René François Walter, baron de Sluze. The curves are defined by the polar equation :

r=\sec\theta+a\cos\theta \,.

cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

, the curves satisfy the

implicit equation 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 ...

(x-1)(x^2+y^2)=ax^2 \,

except that for the implicit form has an

acnode An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are isolated point and hermit point. For example the equation :f(x,y)=y^2+x^2-x^3=0 has an acnode at the origin, because it is ...

not present in polar form. They are

rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...

circular Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation), a document addressed to many destinations ** Government circular, a written statement of government pol ...

cubic plane curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an ...

s. These expressions have an

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, ...

(for ). The point most distant from the asymptote is . is a

crunode In mathematics, a crunode (archaic; from Latin ''crux'' "cross" + ''node'') or node of an algebraic curve is a type of singular point at which the curve intersects itself so that both branches of the curve have distinct tangent lines at the poi ...

for . The area between the curve and the asymptote is, for , :

, a, (1+a/4)\pi \,

while for , the area is :

\left(1-\frac a2\right)\sqrt-a\left(2+\frac a2\right)\arcsin\frac1.

If , the curve will have a loop. The area of the loop is :

\left(2+\frac a2\right)a\arccos\frac1 + \left(1-\frac a2\right)\sqrt.

Four of the family have names of their own: *, line (asymptote to the rest of the family) *,

cissoid of Diocles In geometry, the cissoid of Diocles (; named for Diocles (mathematician), Diocles) is a cubic plane curve notable for the property that it can be used to construct two Geometric mean, mean proportionals to a given ratio. In particular, it can b ...

*, right strophoid *, trisectrix of Maclaurin

References

{{reflist Cubic curves