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Haversine

✖

gives the haversine function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The haversine function is defined by .
  • The argument of haversine is assumed to be in radians. (Multiply by Degree to convert from degrees.)
  • Haversine[z] is the entire function of z with no branch cut discontinuities.
  • Haversine can be evaluated to arbitrary numerical precision.
  • Haversine automatically threads over lists. »
  • Haversine can be used with Interval and CenteredInterval objects. »

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-k3f0cr
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-h40caa
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-kiedlx
Out[1]=1

Series expansion at 0:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-lelxri
Out[1]=1

Scope  (39)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-l274ju
Out[1]=1

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-b0wt9
Out[1]=1

The precision of the output tracks the precision of the input:

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-y7k4a
Out[2]=2

Complex number input:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-bq2c6r
Out[2]=2

Compute the elementwise values of an array using automatic threading:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-thgd2
Out[1]=1

Or compute the matrix Haversine function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-o5jpo
Out[2]=2

Haversine can be used with Interval and CenteredInterval objects:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-h0d6g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-dj6d9x
Out[2]=2

Or compute average-case statistical intervals using Around:

In[3]:=3
✖
https://wolfram.com/xid/0h2q2hqwgu-cw18bq
Out[3]=3

Specific Values  (3)

The values of Haversine at fixed points:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-ft9f33
Out[1]=1

Value at zero:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-m5p0tw
Out[1]=1

Find the first positive extremum of Haversine using Solve:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-f2hrld
Out[1]=1

Substitute in the result:

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-oyn5tf
Out[2]=2

Visualize the result:

In[3]:=3
✖
https://wolfram.com/xid/0h2q2hqwgu-bv6k7w
Out[3]=3

Visualization  (3)

Plot the Haversine function:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-ecj8m7
Out[1]=1

Plot the real part of Haversine(z):

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-bo5grg
Out[1]=1

Plot the imaginary part of Haversine(z):

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-sd9k1
Out[2]=2

Polar plot with r=hav(k phi):

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-epb4bn
Out[1]=1

Function Properties  (13)

Haversine is defined for all real and complex values:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-cl7ele
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-de3irc
Out[2]=2

Haversine achieves all values between zero and one, inclusive, on the reals:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-evf2yr
Out[1]=1

The range for complex values is the whole plane:

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-fphbrc
Out[2]=2

Haversine is periodic with period :

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-ewxrep
Out[1]=1

Expand using ComplexExpand assuming real variables x and y:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-j370ou
Out[1]=1

Haversine has the mirror property hav(TemplateBox[{z}, Conjugate])=TemplateBox[{{hav, (, z, )}}, Conjugate]:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-heoddu
Out[1]=1

Haversine is an analytic function:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-h5x4l2
Out[1]=1

Haversine is neither non-decreasing nor non-increasing:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-g6kynf
Out[1]=1

Haversine is not injective:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-gi38d7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-ctca0g
Out[2]=2

Haversine is not surjective:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-hkqec4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-b1r9xi
Out[2]=2

Haversine is non-negative:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-84dui
Out[1]=1

Haversine has no singularities or discontinuities:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-mdtl3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-mn5jws
Out[2]=2

Haversine is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-kdss3
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-ihw3wi

Differentiation  (3)

The first derivative:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-krpoah
Out[1]=1

Higher derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-nfbe0l
Out[1]=1

Plot the higher derivatives:

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-fxwmfc
Out[2]=2

The formula for the ^(th) derivative:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-cb5zgj
Out[1]=1

Integration  (3)

Compute the indefinite integral using Integrate:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-bponid
Out[1]=1

Verify the anti-derivative:

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-op9yly
Out[2]=2

The definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-b9jw7l
Out[1]=1

More integrals:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-cas
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-hxu52
Out[2]=2

Series Expansions  (4)

Find the Taylor expansion using Series:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-ewr1h8
Out[1]=1

Plots of the first three approximations around :

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-binhar
Out[2]=2

The general term in the series expansion using SeriesCoefficient:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-dznx2j
Out[1]=1

The first-order Fourier series:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-f64drv
Out[1]=1

This happens to be the complete series:

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-x7ucv6
Out[2]=2

The Taylor expansion at a generic point:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-jwxla7
Out[1]=1

Haversine can be applied to a power series:

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-oveup
Out[2]=2

Function Representations  (4)

Haversine can be represent in terms of Sin:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-ffau0o
Out[1]=1

Series representation:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-27x57o
Out[1]=1

Haversine can be represented in terms of MeijerG:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-bwyltp
Out[1]=1

Haversine can be represented as a DifferentialRoot:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-p1kqzy
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

Distance between two points on a sphere:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-g7b3li

Distance between two cities in kilometers (assuming spherical Earth):

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-d7tqsk
In[3]:=3
✖
https://wolfram.com/xid/0h2q2hqwgu-if3q7b
Out[3]=3

Find the distance between the North Pole and the nearest city to it, using the defined function with Haversine:

In[4]:=4
✖
https://wolfram.com/xid/0h2q2hqwgu-4z8kl
Out[4]=4

Properties & Relations  (2)Properties of the function, and connections to other functions

Derivative of haversine function:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-d8dizp
Out[1]=1

Integral of haversine function:

In[2]:=2
✖
https://wolfram.com/xid/0h2q2hqwgu-dhhzby
Out[2]=2

Use FunctionExpand to expand Haversine in terms of standard trigonometric functions:

In[1]:=1
✖
https://wolfram.com/xid/0h2q2hqwgu-en4or3
Out[1]=1
Wolfram Research (2008), Haversine, Wolfram Language function, https://reference.wolfram.com/language/ref/Haversine.html.
✖
Wolfram Research (2008), Haversine, Wolfram Language function, https://reference.wolfram.com/language/ref/Haversine.html.

Text

Wolfram Research (2008), Haversine, Wolfram Language function, https://reference.wolfram.com/language/ref/Haversine.html.

✖
Wolfram Research (2008), Haversine, Wolfram Language function, https://reference.wolfram.com/language/ref/Haversine.html.

CMS

Wolfram Language. 2008. "Haversine." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Haversine.html.

✖
Wolfram Language. 2008. "Haversine." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Haversine.html.

APA

Wolfram Language. (2008). Haversine. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Haversine.html

✖
Wolfram Language. (2008). Haversine. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Haversine.html

BibTeX

@misc{reference.wolfram_2025_haversine, author="Wolfram Research", title="{Haversine}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Haversine.html}", note=[Accessed: 06-June-2025 ]}

✖
@misc{reference.wolfram_2025_haversine, author="Wolfram Research", title="{Haversine}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Haversine.html}", note=[Accessed: 06-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_haversine, organization={Wolfram Research}, title={Haversine}, year={2008}, url={https://reference.wolfram.com/language/ref/Haversine.html}, note=[Accessed: 06-June-2025 ]}

✖
@online{reference.wolfram_2025_haversine, organization={Wolfram Research}, title={Haversine}, year={2008}, url={https://reference.wolfram.com/language/ref/Haversine.html}, note=[Accessed: 06-June-2025 ]}