Limit of the tangent line at a point that tends to infinity
"Asymptotic" redirects here; not to be confused with
Asymptomatic
The graph of a function with a horizontal (
= 0), vertical (
= 0), and oblique asymptote (purple line, given by
= 2
A curve intersecting an asymptote infinitely many times
Look up
asymptote
in Wiktionary, the free dictionary.
In
analytic geometry
, an
asymptote
() of a
curve
is a
straight line
such that the distance between the curve and the line approaches zero as one or both of the
or
coordinates
tends to infinity
. In
projective geometry
and related contexts, an asymptote of a curve is a line which is
tangent
to the curve at a
point at infinity
The word "asymptote" derives from the
Greek
ἀσύμπτωτος (
asumptōtos
), which means "not falling together", from ἀ
priv.
"not" + σύν "together" + πτωτ-ός "fallen".
The term was introduced by
Apollonius of Perga
in his work on
conic sections
, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.
There are three kinds of asymptotes:
horizontal
vertical
and
oblique
. For curves given by the
graph
of a
function
, horizontal asymptotes are horizontal lines that the graph of the function approaches as
tends to
+∞ or −∞.
Vertical asymptotes are vertical lines near which the function grows without bound. An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as
tends to
+∞ or −∞.
More generally, one curve is a
curvilinear asymptote
of another (as opposed to a
linear asymptote
) if the distance between the two curves tends to zero as they tend to infinity, although the term
asymptote
by itself is usually reserved for linear asymptotes.
Asymptotes convey information about the behavior of curves
in the large
, and determining the asymptotes of a function is an important step in sketching its graph.
The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of
asymptotic analysis
{\displaystyle f(x)={\tfrac {1}{x}}}
graphed on
Cartesian coordinates
. The
and
-axis are the asymptotes.
The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem to counter everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width. So if they were to be extended far enough they would seem to merge, at least as far as the eye could discern. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0. Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.
Consider the graph of the function
{\displaystyle f(x)={\frac {1}{x}}}
shown in this section. The coordinates of the points on the curve are of the form
{\displaystyle \left(x,{\frac {1}{x}}\right)}
where x is a number other than 0. For example, the graph contains the points (1, 1), (2, 0.5), (5, 0.2), (10, 0.1), ... As the values of
{\displaystyle x}
become larger and larger, say 100, 1,000, 10,000 ..., putting them far to the right of the illustration, the corresponding values of
{\displaystyle y}
, .01, .001, .0001, ..., become infinitesimal relative to the scale shown. But no matter how large
{\displaystyle x}
becomes, its reciprocal
{\displaystyle {\frac {1}{x}}}
is never 0, so the curve never actually touches the
-axis. Similarly, as the values of
{\displaystyle x}
become smaller and smaller, say .01, .001, .0001, ..., making them infinitesimal relative to the scale shown, the corresponding values of
{\displaystyle y}
, 100, 1,000, 10,000 ..., become larger and larger. So the curve extends further and further upward as it comes closer and closer to the
-axis. Thus, both the
and
-axis are asymptotes of the curve. These ideas are part of the basis of concept of a
limit
in mathematics, and this connection is explained more fully below.
Asymptotes of functions
edit
The asymptotes most commonly encountered in the study of
calculus
are of curves of the form
. These can be computed using
limits
and classified into
horizontal
vertical
and
oblique
asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches as
tends to +∞ or −∞. As the name indicates they are parallel to the
-axis. Vertical asymptotes are vertical lines (perpendicular to the
-axis) near which the function grows without bound. Oblique asymptotes are diagonal lines such that the difference between the curve and the line approaches 0 as
tends to +∞ or −∞.
Vertical asymptotes
edit
The line
is a
vertical asymptote
of the graph of the function
if at least one of the following statements is true:
lim
{\displaystyle \lim _{x\to a^{-}}f(x)=\pm \infty ,}
lim
{\displaystyle \lim _{x\to a^{+}}f(x)=\pm \infty ,}
where
lim
{\displaystyle \lim _{x\to a^{-}}}
is the limit as
approaches the value
from the left (from lesser values), and
lim
{\displaystyle \lim _{x\to a^{+}}}
is the limit as
approaches
from the right.
For example, if ƒ(
) =
/(
–1), the numerator approaches 1 and the denominator approaches 0 as
approaches 1. So
lim
{\displaystyle \lim _{x\to 1^{+}}{\frac {x}{x-1}}=+\infty }
lim
{\displaystyle \lim _{x\to 1^{-}}{\frac {x}{x-1}}=-\infty }
and the curve has a vertical asymptote at
= 1.
The function
) may or may not be defined at
, and its precise value at the point
does not affect the asymptote. For example, for the function
if
if
0.
{\displaystyle f(x)={\begin{cases}{\frac {1}{x}}&{\text{if }}x>0,\\5&{\text{if }}x\leq 0.\end{cases}}}
has a limit of +∞ as
→ 0
) has the vertical asymptote
= 0
, even though
(0) = 5. The graph of this function does intersect the vertical asymptote once, at (0, 5). It is impossible for the graph of a function to intersect a vertical asymptote (or
a vertical line in general
) in more than one point. Moreover, if a function is
continuous
at each point where it is defined, it is impossible that its graph does intersect any vertical asymptote.
A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.
If a function has a vertical asymptote, then it isn't necessarily true that the derivative of the function has a vertical asymptote at the same place. An example is
sin
{\displaystyle f(x)={\tfrac {1}{x}}+\sin({\tfrac {1}{x}})\quad }
at
{\displaystyle \quad x=0}
This function has a vertical asymptote at
{\displaystyle x=0,}
because
lim
lim
sin
{\displaystyle \lim _{x\to 0^{+}}f(x)=\lim _{x\to 0^{+}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=+\infty ,}
and
lim
lim
sin
{\displaystyle \lim _{x\to 0^{-}}f(x)=\lim _{x\to 0^{-}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=-\infty }
The derivative of
{\displaystyle f}
is the function
cos
{\displaystyle f'(x)={\frac {-(\cos({\tfrac {1}{x}})+1)}{x^{2}}}}
For the sequence of points
{\displaystyle x_{n}={\frac {(-1)^{n}}{(2n+1)\pi }},\quad }
for
{\displaystyle \quad n=0,1,2,\ldots }
that approaches
{\displaystyle x=0}
both from the left and from the right, the values
{\displaystyle f'(x_{n})}
are constantly
{\displaystyle 0}
. Therefore, both
one-sided limits
of
{\displaystyle f'}
at
{\displaystyle 0}
can be neither
{\displaystyle +\infty }
nor
{\displaystyle -\infty }
. Hence
{\displaystyle f'(x)}
doesn't have a vertical asymptote at
{\displaystyle x=0}
Horizontal asymptotes
edit
The
arctangent
function has two different asymptotes.
Horizontal asymptotes
are horizontal lines that the graph of the function approaches as
→ ±∞
. The horizontal line
is a horizontal asymptote of the function
) if
lim
{\displaystyle \lim _{x\rightarrow -\infty }f(x)=c}
or
lim
{\displaystyle \lim _{x\rightarrow +\infty }f(x)=c}
In the first case,
) has
as asymptote when
tends to
−∞
, and in the second
) has
as an asymptote as
tends to
+∞
For example, the
arctangent
function satisfies
lim
arctan
{\displaystyle \lim _{x\rightarrow -\infty }\arctan(x)=-{\frac {\pi }{2}}}
and
lim
arctan
{\displaystyle \lim _{x\rightarrow +\infty }\arctan(x)={\frac {\pi }{2}}.}
So the line
= –
/2
is a horizontal asymptote for the arctangent when
tends to
–∞
, and
/2
is a horizontal asymptote for the arctangent when
tends to
+∞
Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. For example, the function
ƒ(
) = 1/(
+1)
has a horizontal asymptote at
= 0 when
tends both to
−∞
and
+∞
because, respectively,
lim
lim
0.
{\displaystyle \lim _{x\to -\infty }{\frac {1}{x^{2}+1}}=\lim _{x\to +\infty }{\frac {1}{x^{2}+1}}=0.}
Other common functions that have one or two horizontal asymptotes include
↦ 1/
(that has an
hyperbola
as it graph), the
Gaussian function
exp
{\displaystyle x\mapsto \exp(-x^{2}),}
the
error function
, and the
logistic function
In the graph of
{\displaystyle f(x)=x+{\tfrac {1}{x}}}
, the
-axis (
= 0) and the line
are both asymptotes.
When a linear asymptote is not parallel to the
- or
-axis, it is called an
oblique asymptote
or
slant asymptote
. A function
) is asymptotic to the straight line
mx
≠ 0) if
lim
or
lim
0.
{\displaystyle \lim _{x\to +\infty }\left[f(x)-(mx+n)\right]=0\,{\mbox{ or }}\lim _{x\to -\infty }\left[f(x)-(mx+n)\right]=0.}
In the first case the line
mx
is an oblique asymptote of
) when
tends to +∞, and in the second case the line
mx
is an oblique asymptote of
) when
tends to −∞.
An example is
) =
+ 1/
, which has the oblique asymptote
(that is
= 1,
= 0) as seen in the limits
lim
{\displaystyle \lim _{x\to \pm \infty }\left[f(x)-x\right]}
lim
{\displaystyle =\lim _{x\to \pm \infty }\left[\left(x+{\frac {1}{x}}\right)-x\right]}
lim
0.
{\displaystyle =\lim _{x\to \pm \infty }{\frac {1}{x}}=0.}
Elementary methods for identifying asymptotes
edit
The asymptotes of many
elementary functions
can be found without the explicit use of limits (although the derivations of such methods typically use limits).
General computation of oblique asymptotes for functions
edit
The oblique asymptote, for the function
), will be given by the equation
mx
. The value for
is computed first and is given by
def
lim
{\displaystyle m\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}f(x)/x}
where
is either
{\displaystyle -\infty }
or
{\displaystyle +\infty }
depending on the case being studied. It is good practice to treat the two cases separately. If this limit doesn't exist then there is no oblique asymptote in that direction.
Having
then the value for
can be computed by
def
lim
{\displaystyle n\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}(f(x)-mx)}
where
should be the same value used before. If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining
exist. Otherwise
mx
is the oblique asymptote of
) as
tends to
For example, the function
) = (2
+ 3
+ 1)/
has
lim
lim
{\displaystyle m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {2x^{2}+3x+1}{x^{2}}}=2}
and then
lim
lim
{\displaystyle n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\left({\frac {2x^{2}+3x+1}{x}}-2x\right)=3}
so that
= 2
+ 3
is the asymptote of
) when
tends to +∞.
The function
) = ln
has
lim
lim
ln
{\displaystyle m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {\ln x}{x}}=0}
and then
lim
lim
ln
{\displaystyle n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\ln x}
, which does not exist.
So
= ln
does not have an asymptote when
tends to +∞.
Asymptotes for rational functions
edit
rational function
has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.
The
degree
of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where:
deg
num
is the degree of the numerator, and
deg
den
is the degree of the denominator.
The cases of horizontal and oblique asymptotes for rational functions
deg
num
- deg
den
Asymptotes in general
Example
Asymptote for example
< 0
{\displaystyle y=0}
{\displaystyle f(x)={\frac {1}{x^{2}+1}}}
{\displaystyle y=0}
= 0
= the ratio of leading coefficients
12
{\displaystyle f(x)={\frac {2x^{2}+7}{3x^{2}+x+12}}}
{\displaystyle y={\frac {2}{3}}}
= 1
= the quotient of the
Euclidean division
of the numerator by the denominator
{\displaystyle f(x)={\frac {2x^{2}+3x+5}{x}}=2x+3+{\frac {5}{x}}}
{\displaystyle y=2x+3}
> 1
none
{\displaystyle f(x)={\frac {2x^{4}}{3x^{2}+1}}}
no linear asymptote, but a
curvilinear asymptote
exists
The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at
= 0, and
= 1, but not at
= 2.
further explanation needed
{\displaystyle f(x)={\frac {x^{2}-5x+6}{x^{3}-3x^{2}+2x}}={\frac {(x-2)(x-3)}{x(x-1)(x-2)}}}
Oblique asymptotes of rational functions
edit
Black: the graph of
{\displaystyle f(x)=(x^{2}+x+1)/(x+1)}
. Red: the asymptote
{\displaystyle y=x}
. Green: difference between the graph and its asymptote for
{\displaystyle x=1,2,3,4,5,6}
When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique (slant) asymptote. The asymptote is the polynomial term after
dividing
the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. For example, consider the function
{\displaystyle f(x)={\frac {x^{2}+x+1}{x+1}}=x+{\frac {1}{x+1}}}
shown to the right. As the value of
increases,
approaches the asymptote
. This is because the other term, 1/(
+1), approaches 0.
If the degree of the numerator is more than 1 larger than the degree of the denominator, and the denominator does not divide the numerator, there will be a nonzero remainder that goes to zero as
increases, but the quotient will not be linear, and the function does not have an oblique asymptote.
Transformations of known functions
edit
If a known function has an asymptote (such as
=0 for
(x)=
), then the translations of it also have an asymptote.
If
is a vertical asymptote of
), then
is a vertical asymptote of
If
is a horizontal asymptote of
), then
is a horizontal asymptote of
)+
If a known function has an asymptote, then the
scaling
of the function also have an asymptote.
If
ax
is an asymptote of
), then
cax
cb
is an asymptote of
cf
For example,
)=
-1
+2 has horizontal asymptote
=0+2=2, and no vertical or oblique asymptotes.
(sec(t), cosec(t)), or x
+ y
= (xy)
, with 2 horizontal and 2 vertical asymptotes
Let
: (
) →
be a
parametric
plane curve, in coordinates
) = (
),
)). Suppose that the curve tends to infinity, that is:
lim
{\displaystyle \lim _{t\rightarrow b}(x^{2}(t)+y^{2}(t))=\infty .}
A line ℓ is an asymptote of
if the distance from the point
) to ℓ tends to zero as
From the definition, only open curves that have some infinite branch can have an asymptote. No closed curve can have an asymptote.
For example, the upper right branch of the curve
= 1/
can be defined parametrically as
= 1/
(where
> 0). First,
→ ∞ as
→ ∞ and the distance from the curve to the
-axis is 1/
which approaches 0 as
→ ∞. Therefore, the
-axis is an asymptote of the curve. Also,
→ ∞ as
→ 0 from the right, and the distance between the curve and the
-axis is
which approaches 0 as
→ 0. So the
-axis is also an asymptote. A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.
Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is
{\displaystyle ax+by+c=0}
then the distance from the point
) = (
),
)) to the line is given by
{\displaystyle {\frac {|ax(t)+by(t)+c|}{\sqrt {a^{2}+b^{2}}}}}
if γ(
) is a change of parameterization then the distance becomes
{\displaystyle {\frac {|ax(\gamma (t))+by(\gamma (t))+c|}{\sqrt {a^{2}+b^{2}}}}}
which tends to zero simultaneously as the previous expression.
An important case is when the curve is the
graph
of a
real function
(a function of one real variable and returning real values). The graph of the function
) is the set of points of the plane with coordinates (
)). For this, a parameterization is
{\displaystyle t\mapsto (t,f(t)).}
This parameterization is to be considered over the open intervals (
), where
can be −∞ and
can be +∞.
An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is
, for some real number
. The non-vertical case has equation
mx
, where
and
{\displaystyle n}
are real numbers. All three types of asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once.
conical spiral
(in red) and its asymptote (in blue).
Asymptotes can also be defined for
space curves
in
Curvilinear asymptotes
edit
+2
+3 is a parabolic asymptote to (
+2
+3
+4)/
x.
Let
: (
) →
be a parametric plane curve, in coordinates
) = (
),
)), and
be another (unparameterized) curve. Suppose, as before, that the curve
tends to infinity. The curve
is a curvilinear asymptote of
if the shortest distance from the point
) to a point on
tends to zero as
. Sometimes
is simply referred to as an asymptote of
, when there is no risk of confusion with linear asymptotes.
For example, the function
{\displaystyle y={\frac {x^{3}+2x^{2}+3x+4}{x}}}
has a curvilinear asymptote
+ 2
+ 3
, which is known as a
parabolic asymptote
because it is a
parabola
rather than a straight line.
10
Asymptotes and curve sketching
edit
Asymptotes are used in procedures of
curve sketching
. An asymptote serves as a guide line to show the behavior of the curve towards infinity.
11
In order to get better approximations of the curve, curvilinear asymptotes have also been used
12
although the term
asymptotic curve
seems to be preferred.
13
cubic curve
the folium of Descartes
(solid) with a single real asymptote (dashed)
The asymptotes of an
algebraic curve
in the
affine plane
are the lines that are tangent to the
projectivized curve
through a
point at infinity
14
For example, one may identify the
asymptotes to the unit hyperbola
in this manner. Asymptotes are often considered only for real curves,
15
although they also make sense when defined in this way for curves over an arbitrary
field
16
A plane curve of degree
intersects its asymptote at most at
−2 other points, by
Bézout's theorem
, as the intersection at infinity is of multiplicity at least two. For a
conic
, there are a pair of lines that do not intersect the conic at any complex point: these are the two asymptotes of the conic.
A plane algebraic curve is defined by an equation of the form
) = 0 where
is a polynomial of degree
{\displaystyle P(x,y)=P_{n}(x,y)+P_{n-1}(x,y)+\cdots +P_{1}(x,y)+P_{0}}
where
is
homogeneous
of degree
. Vanishing of the linear factors of the highest degree term
defines the asymptotes of the curve: setting
, if
) = (
ax
by
−1
, then the line
{\displaystyle Q'_{x}(b,a)x+Q'_{y}(b,a)y+P_{n-1}(b,a)=0}
is an asymptote if
{\displaystyle Q'_{x}(b,a)}
and
{\displaystyle Q'_{y}(b,a)}
are not both zero. If
{\displaystyle Q'_{x}(b,a)=Q'_{y}(b,a)=0}
and
{\displaystyle P_{n-1}(b,a)\neq 0}
, there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called a
parabolic branch
, even when it does not have any parabola that is a curvilinear asymptote. If
{\displaystyle Q'_{x}(b,a)=Q'_{y}(b,a)=P_{n-1}(b,a)=0,}
the curve has a singular point at infinity which may have several asymptotes or parabolic branches.
Over the complex numbers,
splits into linear factors, each of which defines an asymptote (or several for multiple factors). Over the reals,
splits in factors that are linear or quadratic factors. Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have several asymptotes or parabolic branches. It may also occur that such a multiple linear factor corresponds to two
complex conjugate
branches, and does not corresponds to any infinite branch of the real curve. For example, the curve
- 1 = 0
has no real points outside the square
{\displaystyle |x|\leq 1,|y|\leq 1}
, but its highest order term gives the linear factor
with multiplicity 4, leading to the unique asymptote
=0.
Hyperbolas, obtained cutting the same right circular cone with a plane and their asymptotes
The
hyperbola
{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}
has the two asymptotes
{\displaystyle y=\pm {\frac {b}{a}}x.}
The equation for the union of these two lines is
0.
{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=0.}
Similarly, the
hyperboloid
{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1}
is said to have the
asymptotic cone
17
18
0.
{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=0.}
The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.
More generally, consider a surface that has an implicit equation
{\displaystyle P_{d}(x,y,z)+P_{d-2}(x,y,z)+\cdots P_{0}=0,}
where the
{\displaystyle P_{i}}
are
homogeneous polynomials
of degree
{\displaystyle i}
and
{\displaystyle P_{d-1}=0}
. Then the equation
{\displaystyle P_{d}(x,y,z)=0}
defines a
cone
which is centered at the origin. It is called an
asymptotic cone
, because the distance to the cone of a point of the surface tends to zero when the point on the surface tends to infinity.
General references
Specific references
Williamson, Benjamin (1899),
"Asymptotes"
An elementary treatise on the differential calculus
Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the Projective Plane",
Mathematics Magazine
72
(3):
183–
192,
CiteSeerX
10.1.1.502.72
doi
10.2307/2690881
JSTOR
2690881
Oxford English Dictionary
, second edition, 1989.
D.E. Smith,
History of Mathematics, vol 2
Dover (1958) p. 318
Apostol, Tom M.
(1967),
Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra
(2nd ed.), New York:
John Wiley & Sons
ISBN
978-0-471-00005-1
, §4.18.
Reference for section:
"Asymptote"
The Penny Cyclopædia
vol. 2, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London p. 541
Pogorelov, A. V. (1959),
Differential geometry
, Translated from the first Russian ed. by L. F. Boron, Groningen: P. Noordhoff N. V.,
MR
0114163
, §8.
Blasco, Angel; Pérez-Díaz, Sonia (2015).
"Asymptotes of space curves"
Journal of Computational and Applied Mathematics
278
231–
247.
arXiv
1404.6380
doi
10.1016/j.cam.2014.10.013
Fowler, R. H. (1920),
"The elementary differential geometry of plane curves"
Nature
105
(2637), Cambridge, University Press: 321,
Bibcode
1920Natur.105..321G
doi
10.1038/105321a0
hdl
2027/uc1.b4073882
ISBN
0-486-44277-2
CS1 maint: work parameter with ISBN (
link
, p. 89ff.
William Nicholson,
The British enciclopaedia, or dictionary of arts and sciences; comprising an accurate and popular view of the present improved state of human knowledge
, Vol. 5, 1809
Frost, P.
An elementary treatise on curve tracing
(1918)
online
Fowler, R. H.
The elementary differential geometry of plane curves
Cambridge, University Press, 1920, pp 89ff.(
online at archive.org
Frost, P.
An elementary treatise on curve tracing
, 1918, page 5
C.G. Gibson (1998)
Elementary Geometry of Algebraic Curves
, § 12.6 Asymptotes,
Cambridge University Press
ISBN
0-521-64140-3
Coolidge, Julian Lowell (1959),
A treatise on algebraic plane curves
, New York:
Dover Publications
ISBN
0-486-49576-0
MR
0120551
, pp. 40–44.
Kunz, Ernst (2005),
Introduction to plane algebraic curves
, Boston, MA: Birkhäuser Boston,
ISBN
978-0-8176-4381-2
MR
2156630
, p. 121.
L.P. Siceloff, G. Wentworth, D.E. Smith
Analytic geometry
(1922) p. 271
P. Frost
Solid geometry
(1875)
This has a more general treatment of asymptotic surfaces.