LaTeX to CAS translator
Jump to navigation
Jump to search
This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.
The demo-application converts LaTeX functions which directly translate to CAS counterparts.
Functions without explicit CAS support are available for translation via a DRMF package (under development).
The following LaTeX input ...
{\displaystyle a}
... is translated to the CAS output ...
Semantic latex: a
Confidence: 0
Mathematica
Translation: a
Information
Sub Equations
- a
Free variables
- a
Tests
Symbolic
Numeric
SymPy
Translation: a
Information
Sub Equations
- a
Free variables
- a
Tests
Symbolic
Numeric
Maple
Translation: a
Information
Sub Equations
- a
Free variables
- a
Tests
Symbolic
Numeric
Dependency Graph Information
Is part of
- Failed to parse (syntax error): {\displaystyle E = \{P\in \mathbb{R}^2 \,\mid\, |PF_2| +|PF_1 | = 2a \}\}
- Failed to parse (syntax error): {\displaystyle \sqrt{(x - c)^2 + y^2} + \sqrt{(x + c)^2 + y^2} = 2a\}
- Failed to parse (syntax error): {\displaystyle a \ge b > 0 \}
- Failed to parse (syntax error): {\displaystyle \frac{\left(x_1 + su\right)^2}{a^2} + \frac{\left(y_1 + sv\right)^2}{b^2} = 1\ \quad\Longrightarrow\quad 2s\left(\frac{x_1u}{a^2} + \frac{y_1v}{b^2}\right) + s^2\left(\frac{u^2}{a^2} + \frac{v^2}{b^2}\right) = 0\}
- Failed to parse (syntax error): {\displaystyle \frac{\left(x - x_\circ\right)^2}{a^2} + \frac{\left(y - y_\circ\right)^2}{b^2} = 1 \}
- Failed to parse (syntax error): {\displaystyle (x,\, y) = (a \cos t,\, b \sin t),\ 0 \le t < 2\pi\}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\left|PF_1\right|}{\left|Pl_1\right|} = \frac{\left|PF_2\right|}{\left|Pl_2\right|} = e = \frac{c}{a}\}
- Failed to parse (syntax error): {\displaystyle \left|PF_1\right|^2 - \frac{c^2}{a^2}\left|Pl_1\right|^2 = 0\}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{(x - a)^2}{a^2} + \frac{y^2}{b^2} = 1\}
- Failed to parse (syntax error): {\displaystyle \left|\vec c_1\right|^2 + \left|\vec c_2\right|^2 = \cdots = a^2 + b^2\}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{a^2}{b}\}
- Failed to parse (syntax error): {\displaystyle \left(\tfrac{a^2}{c},\, 0\right)\}
- Failed to parse (syntax error): {\displaystyle \kappa = \frac{1}{a^2 b^2}\left(\frac{x^2}{a^4}+\frac{y^2}{b^4}\right)^{-\frac{3}{2}}\}
- Failed to parse (syntax error): {\displaystyle \rho = a^2 b^2 \left(\frac{x^{2}}{a^4} + \frac{y^{2}}{b^4}\right)^\frac{3}{2} = \frac{1}{a^4 b^4} \sqrt{\left(a^4 y^{2} + b^4 x^{2}\right)^3} \}
- Failed to parse (syntax error): {\displaystyle \rho_0 = \frac{b^2}{a}=p\ , \qquad \left(\pm\frac{c^2}{a}\,\bigg|\,0\right)\}
- Failed to parse (syntax error): {\displaystyle \rho_1 = \frac{a^2}{b}\ , \qquad \left(0\,\bigg|\,\pm\frac{c^2}{b}\right)\}
Description
- length of the semi-major axis
- semi-major axis
- distance
- length
- radius
- focus
- parameter name
- semi-major axis of the ellipse
- equation of a standard ellipse
- height
- origin with width
- standard parametric equation
- ellipse
- point
- equation
- eccentricity
- center
- vertex
- parametric representation
- line
- radius of curvature
- semi-minor axis
- circle
- parameter
- point of the ellipse
- center of the ellipse
- axis
- diagram
- left vertex
- semi-latus rectum
- major axis
- ellipse with equation
- expression
- sum
- circumference
- minor axis
- term
- center of curvature
- conjugate diameter
- area
- co-vertex
- angular coordinate
- ellipse 's equation
- representation
- strip of paper
- Metric property
- proof
- trigonometric formula
- quotient
- area formula
- canonical ellipse equation
- directrice
- factor
- height parameter
- interval
- limit
- major/minor semi axis
- new parameter
- perpendicular vector
- radii
- above-mentioned eccentricity
- tangent at a point
- paper strip
- distance between the focus
- Euclidean plane
- locus of point
- set
- set of point
- sum of the distance
- affine transformation of the coordinate
- angle from the positive horizontal axis
- angle of the slope
- arc length
- area of the ellipse
- Bessel
- canonical equation
- canonical form parameter
- canonical form with parametric equation
- case
- close approximation for the circumference
- coordinate
- coordinate equation
- derivative of the standard representation
- distance from a point
- distance from the center
- ellipse 's major axis
- equation of an ellipse
- equation of the tangent
- exact infinite series
- formula
- general case of an ellipse
- general equation 's coefficient
- general form coefficient by the equation
- harmonic mean
- i.e.
- James Ivory
- numerator of these formula
- origin at the center
- other word
- parametric representation of the standard ellipse
- polar coordinate
- polar form
- radius of the large circle
- rational parametric equation of an ellipse
- rotation angle
- series
- Srinivasa Ramanujan
- standard equation of the ellipse
- tangent vector at point
- top half of the ellipse
- vector parametric equation of the tangent
- width
- yield
- constant ratio
- arbitrary point
- origin
- angle of slope
- area of a circle
- arithmetic mean
- article
- auxiliary point
- axis as major axis
- axis of the ellipse
- calculation
- canonical ellipse
- center of the osculating circle
- circle of radius
- circle with midpoint
- closest distance
- complete elliptic integral of the second kind
- concentric circle
- constant area
- curvature
- curve
- different way
- direction
- easy way
- ellipse equation
- ellipse point
- ellipse with equal axis
- ellipse with semi-axis
- elliptical orbit
- end
- endpoint
- endpoint of the ellipse 's major axis
- farthest distance
- generation of point
- geometric mean
- half
- help of trigonometric formula
- incomplete elliptic integral of the second kind
- integral
- intersection point of orthogonal tangent
- intersection point of this line
- inverse function
- line segment
- other focus at angular coordinate
- parallelogram of tangent
- perimeter
- pin
- point on the line
- point towards the center
- polar coordinate with the origin
- principle
- radical by suitable squaring
- radius at apoapsis
- radius at periapsis
- real number
- reference direction
- rhombus with vertex
- section parametric representation
- semi axis
- sign in the denominator
- simple method
- standard ellipse
- standard form of an ellipse
- string
- substrip of length
- suitable coordinate system by an equation
- tangent direction
- tangent line
- term of eccentricity
- tracing point
- triangle
- trigonometric function
- upper bound on the circumference
- upper co-vertex of the ellipse
- upper half of an ellipse
- useful relation
- variable name
- vertical tangent
- method
- area by the same factor
- cases center
- circle with center
- distance of a point
- distance to the focus
- fact
- family of ellipsis
- figure
- focus at the origin
- function
- proof for the pair
- quotient of the distance
- second integral
- stretch
- ellipse at the vertex point
- angle
- Cartesian coordinate
- device
- pair of pole
- pencil at the vertex
- right focus
- sense of the measurement
- standard representation yield
- strip
- substitution
- svg
- triangle inequality
- vector equation
- variation of the paper strip method
- first method
- form
- line 's equation into the ellipse equation
- section
- slope
- standard form
- animation of the variation
- directrix
- directrix of the ellipse
- elementary function
- inner ellipses.gif
- observation that the midpoint
- SliderCrank
- paper strip method
- second method
Complete translation information:
{
"id" : "FORMULA_0cc175b9c0f1b6a831c399e269772661",
"formula" : "a",
"semanticFormula" : "a",
"confidence" : 0.0,
"translations" : {
"Mathematica" : {
"translation" : "a",
"translationInformation" : {
"subEquations" : [ "a" ],
"freeVariables" : [ "a" ]
},
"numericResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"wasAborted" : false,
"crashed" : false,
"testCalculationsGroups" : [ ]
},
"symbolicResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"crashed" : false,
"testCalculationsGroup" : [ ]
}
},
"SymPy" : {
"translation" : "a",
"translationInformation" : {
"subEquations" : [ "a" ],
"freeVariables" : [ "a" ]
},
"numericResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"wasAborted" : false,
"crashed" : false,
"testCalculationsGroups" : [ ]
},
"symbolicResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"crashed" : false,
"testCalculationsGroup" : [ ]
}
},
"Maple" : {
"translation" : "a",
"translationInformation" : {
"subEquations" : [ "a" ],
"freeVariables" : [ "a" ]
},
"numericResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"wasAborted" : false,
"crashed" : false,
"testCalculationsGroups" : [ ]
},
"symbolicResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"crashed" : false,
"testCalculationsGroup" : [ ]
}
}
},
"positions" : [ {
"section" : 0,
"sentence" : 4,
"word" : 15
}, {
"section" : 1,
"sentence" : 3,
"word" : 17
}, {
"section" : 3,
"sentence" : 0,
"word" : 11
}, {
"section" : 3,
"sentence" : 2,
"word" : 21
}, {
"section" : 9,
"sentence" : 4,
"word" : 12
}, {
"section" : 17,
"sentence" : 2,
"word" : 18
}, {
"section" : 24,
"sentence" : 18,
"word" : 39
}, {
"section" : 36,
"sentence" : 0,
"word" : 11
}, {
"section" : 36,
"sentence" : 6,
"word" : 30
}, {
"section" : 37,
"sentence" : 0,
"word" : 11
}, {
"section" : 42,
"sentence" : 8,
"word" : 36
}, {
"section" : 42,
"sentence" : 8,
"word" : 56
} ],
"includes" : [ ],
"isPartOf" : [ "\\frac{x^2}{a^2}+\\frac{y^2}{b^2} = 1", "a\\geq b", "c = \\sqrt{a^2-b^2}", "(x,y) = (a\\cos(t),b\\sin(t)) \\quad \\text{for} \\quad 0\\leq t\\leq 2\\pi", "e = \\frac{c}{a} = \\sqrt{1 - \\frac{b^2}{a^2}}", "2a", "E = \\{P\\in \\mathbb{R}^2 \\,\\mid\\, |PF_2| +|PF_1 | = 2a \\}\\", "e=\\tfrac{c}{a}", "|PF_2| + |PF_1 | = 2a", "V_1 = (a,\\, 0),\\ V_2 = (-a,\\, 0)", "\\sqrt{(x - c)^2 + y^2} + \\sqrt{(x + c)^2 + y^2} = 2a\\", "b^2 = a^2-c^2", "\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1", "y = \\pm\\frac{b}{a}\\sqrt{a^2 - x^2} = \\pm \\sqrt{\\left(a^2 - x^2\\right)\\left(1 - e^2\\right)}", "a,\\; b", "a + ex", "a - ex", "a \\ge b > 0 \\", "\\tfrac{x^2}{a^2} + \\tfrac{y^2}{b^2} = 1", "a < b", "c = \\sqrt{a^2 - b^2}", "e = \\frac{c}{a} = \\sqrt{1 - \\left(\\frac{b}{a}\\right)^2}", "a > b", "a = b", "\\ell = \\frac{b^2}a = a \\left(1 - e^2\\right)", "\\frac{x_1}{a^2}x + \\frac{y_1}{b^2}y = 1", "\\vec x = \\begin{pmatrix}x_1 \\\\ y_1\\end{pmatrix} + s\\begin{pmatrix} \\;\\! -y_1 a^2 \\\\ \\;\\ \\ \\ x_1 b^2\\end{pmatrix}", "\\frac{\\left(x_1 + su\\right)^2}{a^2} + \\frac{\\left(y_1 + sv\\right)^2}{b^2} = 1\\ \\quad\\Longrightarrow\\quad 2s\\left(\\frac{x_1u}{a^2} + \\frac{y_1v}{b^2}\\right) + s^2\\left(\\frac{u^2}{a^2} + \\frac{v^2}{b^2}\\right) = 0\\", "\\frac{x_1}{a^2}u + \\frac{y_1}{b^2}v = 0", "\\begin{pmatrix}\\frac{x_1}{a^2} & \\frac{y_1}{b^2}\\end{pmatrix}", "\\frac{x_1}{a^2}x + \\tfrac{y_1}{b^2}y = k", "\\frac{x_ 1}{a^2}u + \\frac{y_1}{b^2}v \\ne 0", "\\begin{pmatrix} -y_1a^2 & x_1b^2 \\end{pmatrix}", "\\frac{x_1u}{a^2} + \\tfrac{y_1v}{b^2} = 0", "\\frac{\\left(x - x_\\circ\\right)^2}{a^2} + \\frac{\\left(y - y_\\circ\\right)^2}{b^2} = 1 \\", "\\begin{align} A &= a^2 \\sin^2\\theta + b^2 \\cos^2\\theta \\\\ B &= 2\\left(b^2 - a^2\\right) \\sin\\theta \\cos\\theta \\\\ C &= a^2 \\cos^2\\theta + b^2 \\sin^2\\theta \\\\ D &= -2A x_\\circ - B y_\\circ \\\\ E &= - B x_\\circ - 2C y_\\circ \\\\ F &= A x_\\circ^2 + B x_\\circ y_\\circ + C y_\\circ^2 - a^2 b^2.\\end{align}", "\\begin{align} a, b &= \\frac{-\\sqrt{2 \\Big(A E^2 + C D^2 - B D E + (B^2 - 4 A C) F\\Big)\\left((A + C) \\pm \\sqrt{(A - C)^2 + B^2}\\right)}}{B^2 - 4 A C} \\\\ x_\\circ &= \\frac{2CD - BE}{B^2 - 4AC} \\\\[3pt] y_\\circ &= \\frac{2AE - BD}{B^2 - 4AC} \\\\[3pt] \\theta &= \\begin{cases} \\arctan\\left(\\frac{1}{B}\\left(C - A - \\sqrt{(A - C)^2 + B^2}\\right)\\right) & \\text{for } B \\ne 0 \\\\ 0 & \\text{for } B = 0,\\ A < C \\\\ 90^\\circ & \\text{for } B = 0,\\ A > C \\\\ \\end{cases}\\end{align}", "\\tfrac{x^2}{a^2}+\\tfrac{y^2}{b^2} = 1", "(x,\\, y) = (a \\cos t,\\, b \\sin t),\\ 0 \\le t < 2\\pi\\", "\\begin{align} x(u) &= a\\frac{1 - u^2}{u^2 + 1} \\\\ y(u) &= \\frac{2bu}{u^2 + 1}\\end{align}\\;,\\quad -\\infty < u < \\infty\\;", "(-a,\\, 0)", "\\lim_{u \\to \\pm \\infty} (x(u),\\, y(u)) = (-a,\\, 0)\\;", "\\vec x(t) = (a \\cos t,\\, b \\sin t)^\\mathsf{T}", "\\vec x'(t) = (-a\\sin t,\\, b\\cos t)^\\mathsf{T} \\quad \\rightarrow \\quad m = -\\frac{b}{a}\\cot t\\quad \\rightarrow \\quad \\cot t = -\\frac{ma}{b}", "\\cos t = \\frac{\\cot t}{\\pm\\sqrt{1 + \\cot^2t}} = \\frac{-ma}{\\pm\\sqrt{m^2 a^2 + b^2}}\\ ,\\quad\\quad\\sin t = \\frac{1}{\\pm\\sqrt{1 + \\cot^2t}} = \\frac{b}{\\pm\\sqrt{m^2 a^2 + b^2}}", "\\vec c_\\pm(m) = \\left(-\\frac{ma^2}{\\pm\\sqrt{m^2 a^2 + b^2}},\\;\\frac{b^2}{\\pm\\sqrt{m^2a^2 + b^2}}\\right),\\, m \\in \\R", "(\\pm a,\\, 0)", "y = mx \\pm\\sqrt{m^2 a^2 + b^2}\\;", "r(\\theta) = \\frac{ab}{\\sqrt{(b \\cos \\theta)^2 + (a\\sin \\theta)^2}}=\\frac{b}{\\sqrt{1 - (e\\cos\\theta)^2}}", "r(\\theta)=\\frac{a (1-e^2)}{1 \\pm e\\cos\\theta}", "r=\\frac{a (1-e^2)}{1 - e\\cos(\\theta - \\phi)}", "\\ell=a (1-e^2)", "d = \\frac{a^2}{c} = \\frac{a}{e}", "\\frac{\\left|PF_1\\right|}{\\left|Pl_1\\right|} = \\frac{\\left|PF_2\\right|}{\\left|Pl_2\\right|} = e = \\frac{c}{a}\\", "\\left|PF_1\\right|^2 = (x - c)^2 + y^2,\\ \\left|Pl_1\\right|^2 = \\left(x - \\tfrac{a^2}{c}\\right)^2", "y^2 = b^2 - \\tfrac{b^2}{a^2}x^2", "\\left|PF_1\\right|^2 - \\frac{c^2}{a^2}\\left|Pl_1\\right|^2 = 0\\", "a,\\, b", "1 - e^2 = \\tfrac{b^2}{a^2}, \\text{ and }\\ p = \\tfrac{b^2}{a}", "\\frac{(x - a)^2}{a^2} + \\frac{y^2}{b^2} = 1\\", "(a,\\, 0)", "2a = \\left|LF_2\\right| < \\left|QF_2\\right| + \\left|QL\\right| = \\left|QF_2\\right| + \\left|QF_1\\right|", "\\left|QF_2\\right| + \\left|QF_1\\right| > 2a", "c_1^2 + c_2^2 = a^2 + b^2", "\\vec p(t) = (a\\cos t,\\, b\\sin t)", "\\vec c_2 = (-a\\sin t,\\, b\\cos t)^\\mathsf{T}", "\\left|\\vec c_1\\right|^2 + \\left|\\vec c_2\\right|^2 = \\cdots = a^2 + b^2\\", "\\tfrac{x^2}{a^2}+\\tfrac{y^2}{b^2}=1", "x^2+y^2=a^2+b^2", "(a\\cos t,\\, b\\sin t)", "a,b", "a + b", "\\tfrac{a + b}{2}", "a - b", "\\tfrac{b^2}{a}", "\\tfrac{a^2}{b}\\", "C_1 = \\left(a - \\tfrac{b^2}{a}, 0\\right),\\, C_3 = \\left(0, b - \\tfrac{a^2}{b}\\right)", "H = (a,\\, b)", "A = (-a,\\, 2b),\\, B = (a,\\,2b)", "\\tfrac{\\left(x - x_\\circ\\right)^2}{a^2} + \\tfrac{\\left(y - y_\\circ\\right)^2}{b^2} = 1", "{\\color{blue}q} = \\frac{a^2}{b^2} = \\frac{1}{1 - e^2}", "\\left(x - x_\\circ\\right)^2 + {\\color{blue}q}\\, \\left(y - y_\\circ\\right)^2 = a^2", "x_\\circ,\\, y_\\circ,\\, a", "(x - x_\\circ)^2 + {\\color{blue}q}\\, (y - y_\\circ)^2 = a^2", "\\tfrac{x_1x}{a^2} + \\tfrac{y_1y}{b^2} = 1", "\\tfrac{x_1 x}{a^2} + \\tfrac{y_1 y}{b^2} = 1", "\\left(\\tfrac{a^2}{c},\\, 0\\right)\\", "x = \\tfrac{a^2}{c}", "x = -\\tfrac{a^2}{c}", "\\frac{x^2}{a^2}+\\frac{y^2}{b^2}= 1", "\\pi a b", "a/b", "\\pi b^2(a/b) = \\pi a b", "y(x)= b \\sqrt{1 - x^2/a^2}", "x\\in[-a,a]", "[-a,a]", "\\pi a^2", "a^2\\pi\\sqrt{1-e^2}", "C \\,=\\, 4a\\int_0^{\\pi/2}\\sqrt {1 - e^2 \\sin^2\\theta}\\ d\\theta \\,=\\, 4 a \\,E(e)", "e=\\sqrt{1 - b^2/a^2}", "\\begin{align} C &= 2\\pi a \\left[{1 - \\left(\\frac{1}{2}\\right)^2e^2 - \\left(\\frac{1\\cdot 3}{2\\cdot 4}\\right)^2\\frac{e^4}{3} - \\left(\\frac{1\\cdot 3\\cdot 5}{2\\cdot 4\\cdot 6}\\right)^2\\frac{e^6}{5} - \\cdots}\\right] \\\\ &= 2\\pi a \\left[1 - \\sum_{n=1}^\\infty \\left(\\frac{(2n-1)!!}{(2n)!!}\\right)^2 \\frac{e^{2n}}{2n-1}\\right] \\\\ &= -2\\pi a \\sum_{n=0}^\\infty \\left(\\frac{(2n-1)!!}{(2n)!!}\\right)^2 \\frac{e^{2n}}{2n-1},\\end{align}", "h = (a-b)^2 / (a+b)^2", "\\begin{align} C &= \\pi (a+b) \\sum_{n=0}^\\infty \\left(\\frac{(2n-3)!!}{2^n n!}\\right)^2 h^n \\\\ &= \\pi (a+b) \\left[1 + \\frac{h}{4} + \\sum_{n=2}^\\infty \\left(\\frac{(2n-3)!!}{2^n n!}\\right)^2 h^n\\right] \\\\ &= \\pi (a+b) \\left[1 + \\sum_{n=1}^\\infty \\left(\\frac{(2n-1)!!}{2^n n!}\\right)^2 \\frac{h^n}{(2n-1)^2}\\right].\\end{align}", "C \\approx \\pi \\biggl[3(a + b) - \\sqrt{(3a + b)(a + 3b)} \\biggr] = \\pi \\biggl[3(a + b) - \\sqrt{10ab + 3\\left(a^2 + b^2\\right)}\\biggr]", "C\\approx\\pi\\left(a+b\\right)\\left(1+\\frac{3h}{10+\\sqrt{4-3h}}\\right)", "y=b\\sqrt{1-\\frac{x^{2}}{a^{2}}}", "s = -b\\int_{\\arccos \\frac{x_{1}}{a}}^{\\arccos \\frac{x_{2}}{a}} \\sqrt{1-\\left(1-\\frac{a^{2}}{b^{2}}\\right)\\sin^{2}z} \\, dz", "s = -b\\left[E\\left(z \\;\\Biggl|\\; 1 - \\frac{a^{2}}{b^{2}}\\right)\\right]^{\\arccos \\frac{x_{2}}{a}}_{\\arccos \\frac{x_{1}}{a}}", "x^2/a^2 + y^2/b^2 = 1", "\\begin{align} 2\\pi b &\\le C \\le 2\\pi a, \\\\ \\pi (a+b) &\\le C \\le 4(a+b), \\\\ 4\\sqrt{a^2+b^2} &\\le C \\le \\sqrt{2} \\pi \\sqrt{a^2+b^2} .\\end{align}", "2\\pi a", "4\\sqrt{a^2+b^2}", "\\kappa = \\frac{1}{a^2 b^2}\\left(\\frac{x^2}{a^4}+\\frac{y^2}{b^4}\\right)^{-\\frac{3}{2}}\\", "\\rho = a^2 b^2 \\left(\\frac{x^{2}}{a^4} + \\frac{y^{2}}{b^4}\\right)^\\frac{3}{2} = \\frac{1}{a^4 b^4} \\sqrt{\\left(a^4 y^{2} + b^4 x^{2}\\right)^3} \\", "(\\pm a,0)", "\\rho_0 = \\frac{b^2}{a}=p\\ , \\qquad \\left(\\pm\\frac{c^2}{a}\\,\\bigg|\\,0\\right)\\", "\\rho_1 = \\frac{a^2}{b}\\ , \\qquad \\left(0\\,\\bigg|\\,\\pm\\frac{c^2}{b}\\right)\\", "\\begin{align} e &= \\frac{r_a - r_p}{r_a + r_p} = \\frac{r_a - r_p}{2a} \\\\ r_a &= (1 + e)a \\\\ r_p &= (1 - e)a\\end{align}", "r_a", "\\begin{align} a &= \\frac{r_a + r_p}{2} \\\\[2pt] b &= \\sqrt{r_a r_p} \\\\[2pt] \\ell &= \\frac{2}{\\frac{1}{r_a} + \\frac{1}{r_p}} = \\frac{2r_ar_p}{r_a + r_p}\\end{align}" ],
"definiens" : [ {
"definition" : "length of the semi-major axis",
"score" : 0.9122479819684324
}, {
"definition" : "semi-major axis",
"score" : 0.8869384888466118
}, {
"definition" : "distance",
"score" : 0.8465186577350331
}, {
"definition" : "length",
"score" : 0.8465186576477093
}, {
"definition" : "radius",
"score" : 0.8340178737966955
}, {
"definition" : "focus",
"score" : 0.7673244498623755
}, {
"definition" : "parameter name",
"score" : 0.722
}, {
"definition" : "semi-major axis of the ellipse",
"score" : 0.722
}, {
"definition" : "equation of a standard ellipse",
"score" : 0.6798547463189357
}, {
"definition" : "height",
"score" : 0.6798547463189357
}, {
"definition" : "origin with width",
"score" : 0.6798547463189357
}, {
"definition" : "standard parametric equation",
"score" : 0.5886995804672649
}, {
"definition" : "ellipse",
"score" : 0.5230649171097668
}, {
"definition" : "point",
"score" : 0.5000589573182231
}, {
"definition" : "equation",
"score" : 0.49092507339336044
}, {
"definition" : "eccentricity",
"score" : 0.4858515936093988
}, {
"definition" : "center",
"score" : 0.4781729284600878
}, {
"definition" : "vertex",
"score" : 0.47817292846007686
}, {
"definition" : "parametric representation",
"score" : 0.47487475539756296
}, {
"definition" : "line",
"score" : 0.4726978810340824
}, {
"definition" : "radius of curvature",
"score" : 0.47079509830057176
}, {
"definition" : "semi-minor axis",
"score" : 0.46357722582839933
}, {
"definition" : "circle",
"score" : 0.4611534459949592
}, {
"definition" : "parameter",
"score" : 0.44677130157574885
}, {
"definition" : "point of the ellipse",
"score" : 0.4458412414843229
}, {
"definition" : "center of the ellipse",
"score" : 0.43974402504781257
}, {
"definition" : "axis",
"score" : 0.43604830481278595
}, {
"definition" : "diagram",
"score" : 0.4298469245367732
}, {
"definition" : "left vertex",
"score" : 0.42422631924879695
}, {
"definition" : "semi-latus rectum",
"score" : 0.41812910281228655
}, {
"definition" : "major axis",
"score" : 0.41569697587621374
}, {
"definition" : "ellipse with equation",
"score" : 0.4148248297400684
}, {
"definition" : "expression",
"score" : 0.4148248297400684
}, {
"definition" : "sum",
"score" : 0.4148248297400684
}, {
"definition" : "circumference",
"score" : 0.4087276133035581
}, {
"definition" : "minor axis",
"score" : 0.4087276133035581
}, {
"definition" : "term",
"score" : 0.4087276133035581
}, {
"definition" : "center of curvature",
"score" : 0.404973979113224
}, {
"definition" : "conjugate diameter",
"score" : 0.404973979113224
}, {
"definition" : "area",
"score" : 0.4002584833040445
}, {
"definition" : "co-vertex",
"score" : 0.4002584833040445
}, {
"definition" : "angular coordinate",
"score" : 0.38203772243609424
}, {
"definition" : "ellipse 's equation",
"score" : 0.38203772243609424
}, {
"definition" : "representation",
"score" : 0.38203772243609424
}, {
"definition" : "strip of paper",
"score" : 0.38203772243609424
}, {
"definition" : "Metric property",
"score" : 0.3729377917948167
}, {
"definition" : "proof",
"score" : 0.3604245429730373
}, {
"definition" : "trigonometric formula",
"score" : 0.35101160898114886
}, {
"definition" : "quotient",
"score" : 0.34544671201814114
}, {
"definition" : "area formula",
"score" : 0.34544671201814053
}, {
"definition" : "canonical ellipse equation",
"score" : 0.34544671201814053
}, {
"definition" : "directrice",
"score" : 0.34544671201814053
}, {
"definition" : "factor",
"score" : 0.34544671201814053
}, {
"definition" : "height parameter",
"score" : 0.34544671201814053
}, {
"definition" : "interval",
"score" : 0.34544671201814053
}, {
"definition" : "limit",
"score" : 0.34544671201814053
}, {
"definition" : "major/minor semi axis",
"score" : 0.34544671201814053
}, {
"definition" : "new parameter",
"score" : 0.34544671201814053
}, {
"definition" : "perpendicular vector",
"score" : 0.34544671201814053
}, {
"definition" : "radii",
"score" : 0.34544671201814053
}, {
"definition" : "above-mentioned eccentricity",
"score" : 0.34394067119559496
}, {
"definition" : "tangent at a point",
"score" : 0.34068860411922597
}, {
"definition" : "paper strip",
"score" : 0.33653155808959206
}, {
"definition" : "distance between the focus",
"score" : 0.3360452234506229
}, {
"definition" : "Euclidean plane",
"score" : 0.3360452234506229
}, {
"definition" : "locus of point",
"score" : 0.3360452234506229
}, {
"definition" : "set",
"score" : 0.3360452234506229
}, {
"definition" : "set of point",
"score" : 0.3360452234506229
}, {
"definition" : "sum of the distance",
"score" : 0.3360452234506229
}, {
"definition" : "affine transformation of the coordinate",
"score" : 0.33604522250941204
}, {
"definition" : "angle from the positive horizontal axis",
"score" : 0.33604522250941204
}, {
"definition" : "angle of the slope",
"score" : 0.33604522250941204
}, {
"definition" : "arc length",
"score" : 0.33604522250941204
}, {
"definition" : "area of the ellipse",
"score" : 0.33604522250941204
}, {
"definition" : "Bessel",
"score" : 0.33604522250941204
}, {
"definition" : "canonical equation",
"score" : 0.33604522250941204
}, {
"definition" : "canonical form parameter",
"score" : 0.33604522250941204
}, {
"definition" : "canonical form with parametric equation",
"score" : 0.33604522250941204
}, {
"definition" : "case",
"score" : 0.33604522250941204
}, {
"definition" : "close approximation for the circumference",
"score" : 0.33604522250941204
}, {
"definition" : "coordinate",
"score" : 0.33604522250941204
}, {
"definition" : "coordinate equation",
"score" : 0.33604522250941204
}, {
"definition" : "derivative of the standard representation",
"score" : 0.33604522250941204
}, {
"definition" : "distance from a point",
"score" : 0.33604522250941204
}, {
"definition" : "distance from the center",
"score" : 0.33604522250941204
}, {
"definition" : "ellipse 's major axis",
"score" : 0.33604522250941204
}, {
"definition" : "equation of an ellipse",
"score" : 0.33604522250941204
}, {
"definition" : "equation of the tangent",
"score" : 0.33604522250941204
}, {
"definition" : "exact infinite series",
"score" : 0.33604522250941204
}, {
"definition" : "formula",
"score" : 0.33604522250941204
}, {
"definition" : "general case of an ellipse",
"score" : 0.33604522250941204
}, {
"definition" : "general equation 's coefficient",
"score" : 0.33604522250941204
}, {
"definition" : "general form coefficient by the equation",
"score" : 0.33604522250941204
}, {
"definition" : "harmonic mean",
"score" : 0.33604522250941204
}, {
"definition" : "i.e.",
"score" : 0.33604522250941204
}, {
"definition" : "James Ivory",
"score" : 0.33604522250941204
}, {
"definition" : "numerator of these formula",
"score" : 0.33604522250941204
}, {
"definition" : "origin at the center",
"score" : 0.33604522250941204
}, {
"definition" : "other word",
"score" : 0.33604522250941204
}, {
"definition" : "parametric representation of the standard ellipse",
"score" : 0.33604522250941204
}, {
"definition" : "polar coordinate",
"score" : 0.33604522250941204
}, {
"definition" : "polar form",
"score" : 0.33604522250941204
}, {
"definition" : "radius of the large circle",
"score" : 0.33604522250941204
}, {
"definition" : "rational parametric equation of an ellipse",
"score" : 0.33604522250941204
}, {
"definition" : "rotation angle",
"score" : 0.33604522250941204
}, {
"definition" : "series",
"score" : 0.33604522250941204
}, {
"definition" : "Srinivasa Ramanujan",
"score" : 0.33604522250941204
}, {
"definition" : "standard equation of the ellipse",
"score" : 0.33604522250941204
}, {
"definition" : "tangent vector at point",
"score" : 0.33604522250941204
}, {
"definition" : "top half of the ellipse",
"score" : 0.33604522250941204
}, {
"definition" : "vector parametric equation of the tangent",
"score" : 0.33604522250941204
}, {
"definition" : "width",
"score" : 0.33604522250941204
}, {
"definition" : "yield",
"score" : 0.33604522250941204
}, {
"definition" : "constant ratio",
"score" : 0.31725078032813114
}, {
"definition" : "arbitrary point",
"score" : 0.3111776686501416
}, {
"definition" : "origin",
"score" : 0.3111776686501416
}, {
"definition" : "angle of slope",
"score" : 0.30935533164194823
}, {
"definition" : "area of a circle",
"score" : 0.30935533164194823
}, {
"definition" : "arithmetic mean",
"score" : 0.30935533164194823
}, {
"definition" : "article",
"score" : 0.30935533164194823
}, {
"definition" : "auxiliary point",
"score" : 0.30935533164194823
}, {
"definition" : "axis as major axis",
"score" : 0.30935533164194823
}, {
"definition" : "axis of the ellipse",
"score" : 0.30935533164194823
}, {
"definition" : "calculation",
"score" : 0.30935533164194823
}, {
"definition" : "canonical ellipse",
"score" : 0.30935533164194823
}, {
"definition" : "center of the osculating circle",
"score" : 0.30935533164194823
}, {
"definition" : "circle of radius",
"score" : 0.30935533164194823
}, {
"definition" : "circle with midpoint",
"score" : 0.30935533164194823
}, {
"definition" : "closest distance",
"score" : 0.30935533164194823
}, {
"definition" : "complete elliptic integral of the second kind",
"score" : 0.30935533164194823
}, {
"definition" : "concentric circle",
"score" : 0.30935533164194823
}, {
"definition" : "constant area",
"score" : 0.30935533164194823
}, {
"definition" : "curvature",
"score" : 0.30935533164194823
}, {
"definition" : "curve",
"score" : 0.30935533164194823
}, {
"definition" : "different way",
"score" : 0.30935533164194823
}, {
"definition" : "direction",
"score" : 0.30935533164194823
}, {
"definition" : "easy way",
"score" : 0.30935533164194823
}, {
"definition" : "ellipse equation",
"score" : 0.30935533164194823
}, {
"definition" : "ellipse point",
"score" : 0.30935533164194823
}, {
"definition" : "ellipse with equal axis",
"score" : 0.30935533164194823
}, {
"definition" : "ellipse with semi-axis",
"score" : 0.30935533164194823
}, {
"definition" : "elliptical orbit",
"score" : 0.30935533164194823
}, {
"definition" : "end",
"score" : 0.30935533164194823
}, {
"definition" : "endpoint",
"score" : 0.30935533164194823
}, {
"definition" : "endpoint of the ellipse 's major axis",
"score" : 0.30935533164194823
}, {
"definition" : "farthest distance",
"score" : 0.30935533164194823
}, {
"definition" : "generation of point",
"score" : 0.30935533164194823
}, {
"definition" : "geometric mean",
"score" : 0.30935533164194823
}, {
"definition" : "half",
"score" : 0.30935533164194823
}, {
"definition" : "help of trigonometric formula",
"score" : 0.30935533164194823
}, {
"definition" : "incomplete elliptic integral of the second kind",
"score" : 0.30935533164194823
}, {
"definition" : "integral",
"score" : 0.30935533164194823
}, {
"definition" : "intersection point of orthogonal tangent",
"score" : 0.30935533164194823
}, {
"definition" : "intersection point of this line",
"score" : 0.30935533164194823
}, {
"definition" : "inverse function",
"score" : 0.30935533164194823
}, {
"definition" : "line segment",
"score" : 0.30935533164194823
}, {
"definition" : "other focus at angular coordinate",
"score" : 0.30935533164194823
}, {
"definition" : "parallelogram of tangent",
"score" : 0.30935533164194823
}, {
"definition" : "perimeter",
"score" : 0.30935533164194823
}, {
"definition" : "pin",
"score" : 0.30935533164194823
}, {
"definition" : "point on the line",
"score" : 0.30935533164194823
}, {
"definition" : "point towards the center",
"score" : 0.30935533164194823
}, {
"definition" : "polar coordinate with the origin",
"score" : 0.30935533164194823
}, {
"definition" : "principle",
"score" : 0.30935533164194823
}, {
"definition" : "radical by suitable squaring",
"score" : 0.30935533164194823
}, {
"definition" : "radius at apoapsis",
"score" : 0.30935533164194823
}, {
"definition" : "radius at periapsis",
"score" : 0.30935533164194823
}, {
"definition" : "real number",
"score" : 0.30935533164194823
}, {
"definition" : "reference direction",
"score" : 0.30935533164194823
}, {
"definition" : "rhombus with vertex",
"score" : 0.30935533164194823
}, {
"definition" : "section parametric representation",
"score" : 0.30935533164194823
}, {
"definition" : "semi axis",
"score" : 0.30935533164194823
}, {
"definition" : "sign in the denominator",
"score" : 0.30935533164194823
}, {
"definition" : "simple method",
"score" : 0.30935533164194823
}, {
"definition" : "standard ellipse",
"score" : 0.30935533164194823
}, {
"definition" : "standard form of an ellipse",
"score" : 0.30935533164194823
}, {
"definition" : "string",
"score" : 0.30935533164194823
}, {
"definition" : "substrip of length",
"score" : 0.30935533164194823
}, {
"definition" : "suitable coordinate system by an equation",
"score" : 0.30935533164194823
}, {
"definition" : "tangent direction",
"score" : 0.30935533164194823
}, {
"definition" : "tangent line",
"score" : 0.30935533164194823
}, {
"definition" : "term of eccentricity",
"score" : 0.30935533164194823
}, {
"definition" : "tracing point",
"score" : 0.30935533164194823
}, {
"definition" : "triangle",
"score" : 0.30935533164194823
}, {
"definition" : "trigonometric function",
"score" : 0.30935533164194823
}, {
"definition" : "upper bound on the circumference",
"score" : 0.30935533164194823
}, {
"definition" : "upper co-vertex of the ellipse",
"score" : 0.30935533164194823
}, {
"definition" : "upper half of an ellipse",
"score" : 0.30935533164194823
}, {
"definition" : "useful relation",
"score" : 0.30935533164194823
}, {
"definition" : "variable name",
"score" : 0.30935533164194823
}, {
"definition" : "vertical tangent",
"score" : 0.30935533164194823
}, {
"definition" : "method",
"score" : 0.2808939985231694
}, {
"definition" : "area by the same factor",
"score" : 0.269521391310941
}, {
"definition" : "cases center",
"score" : 0.269521391310941
}, {
"definition" : "circle with center",
"score" : 0.269521391310941
}, {
"definition" : "distance of a point",
"score" : 0.269521391310941
}, {
"definition" : "distance to the focus",
"score" : 0.269521391310941
}, {
"definition" : "fact",
"score" : 0.269521391310941
}, {
"definition" : "family of ellipsis",
"score" : 0.269521391310941
}, {
"definition" : "figure",
"score" : 0.269521391310941
}, {
"definition" : "focus at the origin",
"score" : 0.269521391310941
}, {
"definition" : "function",
"score" : 0.269521391310941
}, {
"definition" : "proof for the pair",
"score" : 0.269521391310941
}, {
"definition" : "quotient of the distance",
"score" : 0.269521391310941
}, {
"definition" : "second integral",
"score" : 0.269521391310941
}, {
"definition" : "stretch",
"score" : 0.269521391310941
}, {
"definition" : "ellipse at the vertex point",
"score" : 0.22226421155186268
}, {
"definition" : "angle",
"score" : 0.22226421155157317
}, {
"definition" : "Cartesian coordinate",
"score" : 0.22226421155157317
}, {
"definition" : "device",
"score" : 0.22226421155157317
}, {
"definition" : "pair of pole",
"score" : 0.22226421155157317
}, {
"definition" : "pencil at the vertex",
"score" : 0.22226421155157317
}, {
"definition" : "right focus",
"score" : 0.22226421155157317
}, {
"definition" : "sense of the measurement",
"score" : 0.22226421155157317
}, {
"definition" : "standard representation yield",
"score" : 0.22226421155157317
}, {
"definition" : "strip",
"score" : 0.22226421155157317
}, {
"definition" : "substitution",
"score" : 0.22226421155157317
}, {
"definition" : "svg",
"score" : 0.22226421155157317
}, {
"definition" : "triangle inequality",
"score" : 0.22226421155157317
}, {
"definition" : "vector equation",
"score" : 0.22226421155157317
}, {
"definition" : "variation of the paper strip method",
"score" : 0.20487522259065638
}, {
"definition" : "first method",
"score" : 0.17354195011337867
}, {
"definition" : "form",
"score" : 0.17354195011337867
}, {
"definition" : "line 's equation into the ellipse equation",
"score" : 0.17354195011337867
}, {
"definition" : "section",
"score" : 0.17354195011337867
}, {
"definition" : "slope",
"score" : 0.17354195011337867
}, {
"definition" : "standard form",
"score" : 0.17354195011337867
}, {
"definition" : "animation of the variation",
"score" : 0.12835413759959
}, {
"definition" : "directrix",
"score" : 0.12835413759959
}, {
"definition" : "directrix of the ellipse",
"score" : 0.12835413759959
}, {
"definition" : "elementary function",
"score" : 0.12835413759959
}, {
"definition" : "inner ellipses.gif",
"score" : 0.12835413759959
}, {
"definition" : "observation that the midpoint",
"score" : 0.12835413759959
}, {
"definition" : "SliderCrank",
"score" : 0.12835413759959
}, {
"definition" : "paper strip method",
"score" : 0.0599388741858322
}, {
"definition" : "second method",
"score" : 0.03802707603753016
} ]
}