§8 Important plane curve table
[ cubic curve ]
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_{}
_{}
(a >0 , b<0 ,Δ>0) (a >0 , b<0 ,Δ=0) (a) (b)
These curves are all symmetrical about a straight line_{} 
The curve consists of two Asymptote y = a and x = 0 The intersection of the curve with y = a _{} The intersection of the curve and the x axis _{} Extreme point _{} inflection point _{}
( a ) discontinuity point _{} maximum point _{} asymptote _ _{} （b） discontinuity point _{} asymptote _ _{} （c） maximum point _{} inflection point _{} The slopes at these two points are _{} Asymptote y = 0

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_{}
[ Note ] a , b are the two roots of the equation , and let_{}_{}

（a） Discontinuous point x = a , x = b Asymptote y = 0 and x = a , x = b （b） Discontinuous point x = a , x = b maximum point _{} minimum point _{} Inflection point C Asymptote y = 0 and x = a , x = b ( c ) Discontinuous point x = a , x = b maximum point _{} minimum point _{} Inflection point C Asymptote y = 0 and x = a , x = b （d） discontinuity point _{} maximum point _{} Inflection point C Asymptote y = 0 and _{} （e） discontinuity point _{} minimum point _{} Inflection point C Asymptote y = 0 and _{} （f） maximum point _{} minimum point _{} Inflection point C , D , E three points Asymptote y = 0 
The graph of the above cubic curve only lists the cases where a > 0. For a < 0 , the division curve (when a > 0 , the asymptote is above the x axis, and when a < 0 , the asymptote is below the x  axis) ), generally after making appropriate changes, the curves with a > 0 are symmetrical about the x axis . For example , when a < 0 , the two curves: and are symmetrical about the x axis, while the latter has a x ^{2} coefficient ._{}_{}_{}^{}_{}
[ Parabolic Curve ]
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_{}
_{}
_{} 
(a) , where n is even:_{} y changes from to_{}_{} Odd number of extreme points ( 1 ~ n  1 ) Even number of inflection points ( 0 ~ n  2 )
(b) , where n is odd_{} y changes from to_{}_{} Even number of extreme points ( 0 ~ n  1 ) Odd number of inflection points ( 1 ~ n  2 )
The intersections A _{1} , A _{2} , A _{3} ( or an intersection A _{1} ) of the curve and the x axis are the intersections of the real roots of the equation and the y axis _{}_{}_{}_{}_{}_{} C , D at the extreme point ( C takes a positive sign, D takes a negative sign)_{} _{} inflection point it is a curve _{} the center of symmetry of , the slope of the tangent at this point is_{}

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(a) (b) 
(a) , where n is even:_{} Vertex (same extreme point) O (0, 0) The curve is symmetrical about the y axis (b) , where n is odd:_{} Inflection point O (0, 0) The curve is symmetrical about the origin

_{}( m , n are two coprime integers)
_{}

n is even m odd 
n is odd meven _ 
n is odd m odd 
Tangent case 
m>n
m<n
Symmetric case 
Symmetry about the x axis 
Symmetry about the y axis 
Symmetry about the origin 
Tangent to the x axis at the origin
Tangent to the y axis at the origin 
Equations and Graphics 
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[ Hyperbolic Curve ] _{}
_{}
[ Halfcubic parabola ] _{}
[ Kei tongue line ] _{}
[ Cartesian lobe line ] _{} or_{}
[ vine leaf line ] _{} or_{} or_{}
[ ring rope ] _{} or_{} or_{}
[ Nagomedus Clam Line ] _{} or_{} or (positive sign for outer branch, negative sign for inner branch)_{}

( a ) n is even: Discontinuity point O ( 0, 0 ) Asymptote y = 0 and x = 0 The curve is symmetrical about the y axis ( b ) n is odd: The curve is symmetrical about the origin
( a ) , where n is even and m is odd:_{} Discontinuity point O ( 0, 0 ) Asymptote y = 0 and x = 0 The curve is symmetrical about x ( b ) , where n is odd and m is even:_{} The curve is symmetric about y ( c ) , where n is odd and m is odd:_{} The curve is symmetrical about the origin
The point at which the cusp is tangent to the x axis _{} Radius of curvature _{} arc length _{}
the maximum point at which the radius of curvature is _{}_{} The inflection point, the slopes of the tangent lines at these two points are _{}_{} asymptote _ _{} area between curve and asymptote _{}
The node at which it is tangent to the x and y axes and has a radius of curvature of _{}_{} vertex _{} asymptote _{} The area enclosed by the trap_{} area between curve and asymptote _{}
The curve is the locus of the point M made ( P is the intersection of the mother circle of diameter a and OQ )_{} The cusp at which the curve is tangent to the x  axis _{} asymptote _{} area between curve and asymptote_{}
The curve is the locus of points M _{1} , M _{2 that make }PM _{1} = PM _{2} = OP ( P is a point on the y axis, M _{1} , M _{2} are on the ray passing through the two points A , P )_{} _{}_{}_{} vertex A ( a , 0) Node O (0 , 0) Asymptote x =  a The area enclosed by the trap _{} area between curve and asymptote _{}
The curve is the locus of points M _{1} , M _{2 such that }OM _{1} = OP + b , OM _{2} = OP  b ( referred to as outer branch ( right ) and inner branch ( left ), respectively) _{}_{} Outer branch Vertex A ( a + b , 0) Inflection points B , C whose abscissas are equal to the largest root of the equation x ^{3} – 3 a ^{2 }x + 2 a ( a ^{2} – b ^{2} ) = 0 ^{}^{}^{}^{} Inner branch Vertex D ( a  b , 0) Inflection points ( when a > b ) E , F , whose abscissas are equal to the second positive root of the equation x ^{3} – 3 a ^{2 }x + 2 a ( a ^{2} – b ^{2} ) = 0 ^{}^{}^{}^{} cusp ( when a = b ) O (0, 0) Node ( when a < b ) O (0, 0) Asymptotes of the inner and outer branches x = a 
[ Pascal Snail ]
_{}
or _{}
or _{}
The curve is the locus of point M such that OM = OP ± b (point P is on a circle of diameter a )
Vertices A _{k} , B _{k} ( a ± b , 0)( k =1, 2, 3, 4, 5) , B _{2} coincides with the origin _{}_{} _{}
The node ( when b < a ) is O (0, 0) , the slope of the tangent at this point is , and the radius of curvature of this point is _{}_{}
cusp ( when b = a ) O (0, 0)
Outliers ( when b > a ) O (0, 0)
There are 4 extreme points when b < a , and 2 when b 3 a : when b changes from 0 to ￥ , all extreme points form a vineleaf line _{}_{}
Inflection point ( when a < b < 2 a )_{}
Tangent point of the double tangent ( when b < 2 a ):
_{}
The area enclosed by these tangent points on the circle r =  a cos j by the snail line
_{}
(When b < a , the area of the inner circle is calculated twice)
[ Note ] When b = a , it is the heart line .
[ Cassini Oval Line ]
_{}
or_{}
The curve is the locus of point M such that MF _{1} × MF _{2} = a ^{2} ( F _{1} , F _{2} are fixed focus, F _{1 }F _{2} = 2 c , a is constant) ._{}_{}_{}_{}
vertex _{}
_{}
Extreme point _{}
_{}
or_{}
When a changes from 0 to 0 , all extreme points form a circle (radius c )_{}
inflection point _{}
_{}
in
_{}
or
_{}
When a changes from c to , all inflection points form a double kinks_{}
[ Note ] When a = c , it is a double dash .
[ heart line ] _{} or _{} or _{}
[ double twist ] _{} or _{}
[ Ordinary Trochoid (Cycloid) ] _{} or _{}
[ Long (or short) spoke trochoids (trochoids) ] _{}
Long axis ( λ > 1)
Short axis ( λ < 1)
[ Hypotrochle (Epicycloid) ] _{} ( a is the radius of the fixed circle, b is the radius of the moving circle, t = D COx )

(i) It is the locus of point M such that OM = OP ± a ( a is the diameter of the circle, P is a point on the circumference) ( ii ) It is a special case of epitrochoid (moving and definite circles have equal diameters) cusp O (0 , 0) Vertex A (2 a , 0) Extreme point _{} Tangent point of double tangent _{} Curve length L = 8 a area _{}
(i) It is the locus of point M such that MF _{1} × MF _{2} = a ^{2} ( OF _{1} = a ) _{}^{}_{} (ii) It is the locus of point M such that OM = PQ ( P , Q on _{a} circle with center F1 and radius )_{}_{} Node (same as the inflection point ) O (0, 0) at which the slope of the tangent is ± 1 vertex _{} Extreme point _{} Radius of curvature _{} Double knot area S = 2 a ^{2} ^{}
The curve is the trajectory drawn by a point M on the circumference when a circle rolls along the x axis without sliding (the radius of the circle is a ) Period T = 2 p a Extreme point _{} Radius of curvature _{} The involute is a cycloid (dotted line in the figure) arch length _{} area _{}
A curve is a track that a circle rolls along the x axis without sliding pairs, a point M outside the circle (or a point N inside the circle ) (the radius of the circle is a ) Period T = 2 p a node _ _{} _{} inflection point _{} maximum point _{} minimum point _{} Radius of curvature _{} The radius of curvature corresponding to the extreme point ( when l < 1 ) is_{}
A curve is the trajectory drawn by a point M on the circumference when one circumference rolls along the outside of another circumference without sliding, and the shape of the curve is determined by the value of_{} (i) When m = 1 , the curve is the heart line (ii) When m is an integer, the curve consists of m branches. After the moving point M traces the m branches (that is, the moving circle circles around the fixed circle), it returns to the starting position. (iii) When m is a fraction ( , g , h are coprime integers), the curve is composed of g branches. After the moving point M traces the g branch (that is, the moving circle revolves around the fixed circle h ), it returns to the starting position_{} (iv) When m is an irrational number, there are infinitely many branches, and the moving point M cannot return to the starting position sharp point _{}
Vertex (where k is an integer, when m is an integer, ; then , ; when m is an irrational number, ) _{} _{}_{}_{}_{} Curve length (one piece)_{} Radius of curvature _{} Area of sector A _{1 }B _{1 }A _{2 }A _{1} (excluding the area of the fixed circle) _{} 
[ Hypotrochle (hypocycloid) ]
_{}
( a is the radius of the fixed circle, b is the radius of the moving circle, t = D COx )
A curve is the trajectory traced by a point M on a circle when one circle rolls along the interior of another circle without sliding .
The coordinates of the cusp, vertex, arc length, radius of curvature and area of the epitrochle are the same as those of the epitrochle, just replace " + b " with " b " . It is always greater than 1 , especially , when m = 4 , there are 4 curves , called starshaped lines, and the equation is_{}
_{}or_{}
Full curve length L = 6 a
The area enclosed by the curve _{}
[ Long (or short) spoke epitrochoids (exotrochoids) ]
_{}( a is the radius of the fixed circle, b is the radius of the moving circle)
A curve is the trajectory traced by a point M outside ( or inside ) a circle when one circle rolls along the outside of another circle without sliding .
[ Long ( or short ) spoke hypotrochle (hypotrochoid) ]
_{}( a is the radius of the fixed circle, b is the radius of the moving circle)
long spokes short spokes_{} _{}
A curve is the trajectory traced by a point M outside ( or inside ) a circle when one circle rolls along the inside of another circle without sliding . In particular, when a = 2 b , the long and short spoke hypotrochle is an ellipse; When a = b , it is a Pascal thread .
Equations and Graphics 
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[ Archimedes Spiral ] _{}

The curve is the trajectory drawn by a moving point when it moves along a ray at a constant speed , and the ray rotates around the pole O at a constant angular velocity . The curve consists of two curves, which are symmetrical about the x axis ._{}_{} (in the equation , )_{}_{} The ray and the curve passing through the pole are equally spaced at A _{0} , A _{1} , A _{2} , L , and they are equally spaced ( ) _{}_{}_{}_{} arc length _{}_{} Radius of curvature _{} Area of sector M _{1 }OM _{2} _{} 
[ Logarithmic spiral (equiangular spiral) ] _{}

The angle of intersection of the curve and all rays passing through the pole is equal (a)(k=cota) , when the curve rotates clockwise around the pole and tends to the pole_{} The ray and the curve passing through the pole in the proportionality intersect at L , A _{1} , A _{0} , A _{1} , L , then L , , OA _{0} , OA _{1} , L , each line segment is a proportional series (common ratio is ) _{}_{}_{}_{}_{}_{}_{} arc length _{} _{} Radius of curvature _{} [ Note ] At that time , it was a circle_{} 
[ Hybolic spiral ( inverse spiral )]
_{}
Asymptotic point pole O ( at that time ) _{}
Asymptote y = a
Radius of curvature _{}
Area of sector M _{1 }OM _{2} _{}
The curve consists of two legs, which are symmetrical about the y axis
[ Chain Helix ]
_{}
The curve is the locus of the point M that keeps the area of the circular sector OMN constant when N moves on the x axis_{}
Asymptotic point pole O ( at that time ) _{}
Asymptote x  axis ( at the time ) _{}
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[ circular involute ] _{}( a is the radius of the circle, t = COx )_{}

cusp A ( a ,,0) Intersection with the x axis B ( ,0) _{} (where t _{0} is the root of the equation t = tan t ) arc length _{} Radius of curvature _{} The center of curvature C is on the circumference The curve consists of two legs, which are symmetrical about the x axis 
[ clothoid ] _{}

Inflection point O (0,0) Asymptotic point A ( ) _{} B( )_{} arc length _{} The curve is symmetrical about the origin 
Equations and Graphics 
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[ catenary ] _{}

vertex A (0, a ) arc length _{} The area of the curvedsided trapezoid OAMP _{} Radius of curvature _{} The curve is symmetrical about the y axis and lies above the parabola y = a + (dashed line in the figure)_{} [ Note ] Suspend a soft and inextensible heavy rope from the Two points, we get the shape of the catenary

[ tracing line ] x=aArch_{}_{} or x=aln_{}

cusp A (0, a ) at which point is tangent to the y axis arc length = a ln _{}_{} Radius of curvature R = a cot _{} The curve is symmetrical about the y axis, and it is the involute of the catenary, which starts from the vertex A , and the distance from the intersection of the tangent to the x axis to the tangent is a constant . [ Note ] the length of a soft and inextensible rope One end is tied to a mass point M , and the other end P moves along the axis x , Then the point M is drawn into a tractor shape 
[ Rose line ]_{}
Equations and Graphics 
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[ Probability Curve ] y=a (a>0, k>0)_{}

Vertex (Homomax) A (0, a ) Inflection point B , C _{} The slope of the tangent at this point is_{} The area between the curve and the x axis_{} The curve is symmetrical about the y axis

[ Standard normal distribution curve (Gaussian curve) ] y =_{}

Vertex A (0, 0.3989) _{} Inflection point B , C ( ) _{} The area between the curve and the x axis is 1 The curve is symmetrical about the y axis 
[ General normal distribution curve ] y =_{}
[ Damp vibration curve ] y=A (A>0)_{}

vertex (homomaximum point) A(m,)_{} Inflection point B , C ( m ) _{} The area between the curve and the x axis is 1 The curve is symmetrical about the line x = m
Intersection with the x axis B_{k} (k=1,2L)_{} Intersection with y axis C (0, A sin _{0} )_{}_{} The abscissa of the extreme point _{Ak is} _{} _{}(in the formula tan )_{} _{The abscissa} of the inflection point Dk is _{} _{}(in the formula tan )_{} Curve and Two Exponential Curves y = tangent, tangent point_{} P_{k}_{} 