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| [[Image:SnellPotFigure1.png|right]]
| | Surely the second option would be more beneficial for any website. It is very easy to customize plugins according to the needs of a particular business. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>Creating a website from scratch can be such a pain. Should you loved this short article and you wish to receive more info relating to [http://miniURL.fouiner.info/wordpress_backup_plugin_730027 wordpress backup] i implore you to visit the web page. If a newbie missed a certain part of the video then they could always rewind. A Wordpress plugin is a software that you can install into your Wordpress site. Now, I want to anxiety that not every single query will be answered. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available. <br><br>Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. You've got invested a great cope of time developing and producing up the topic substance. The first thing you need to do is to choose the right web hosting plan. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>The disadvantage is it requires a considerable amount of time to set every thing up. I didn't straight consider near it solon than one distance, I got the Popup Ascendancy plugin and it's up and lengthways, likely you make seen it today when you visited our blog, and I yet customize it to fit our Thesis Wound which gives it a rattling uncomparable visage and search than any different popup you know seen before on any added journal, I hump arrogated asset of one of it's quatern themes to make our own. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. The most important plugins you will need are All-in-One SEO Pack, some social bookmarking plugin, a Feedburner plugin and an RSS sign up button. Where from they are coming, which types of posts are getting top traffic and many more. <br><br>Yet, overall, less than 1% of websites presently have mobile versions of their websites. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. This allows updates to be sent anyone who wants them via an RSS reader or directly to their email. In addition, Word - Press design integration is also possible. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities. |
| The '''Snellius–Pothenot problem''' is a problem in planar [[surveying]]. Given three known points A, B and C, an observer at an unknown point P observes that the segment AC subtends an angle <math>\alpha</math> and the segment CB subtends an angle <math>\beta</math>; the problem is to determine the position of the point P. (See figure; the point denoted C is between A and B as seen from P).
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| Since it involves the observation of known points from an unknown point, the problem is an example of [[Resection (orientation)|resection]]. Historically it was first studied by [[Willebrord Snellius|Snellius]], who found a solution around 1615.
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| ==Formulating the equations==
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| '''First equation'''
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| Denoting the (unknown) angles ''CAP'' as ''x'' and ''CBP'' as ''y'' we get:
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| :<math>x+y = 2 \pi - \alpha - \beta - C</math>
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| by using the sum of the angles formula for the [[quadrilateral]] ''PACB''. The variable ''C'' represents the (known) internal angle in this quadrilateral at point ''C''. (Note that in the case where the points ''C'' and ''P'' are on the same side of the line ''AB'', the angle C will be greater than <math>\pi</math>).
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| '''Second equation'''
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| Applying the [[law of sines]] in triangles PAC and PBC we can express PC in two different ways:
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| :<math>\frac{\rm{AC} \sin x }{\sin \alpha} = \rm{PC} = \frac{\rm{BC} \sin y}{\sin \beta}.</math>
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| A useful trick at this point is to define an auxiliary angle <math>\phi</math> such that
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| :<math>\tan \phi = \frac{\rm{BC} \sin \alpha}{\rm{AC} \sin \beta}.</math>
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| (A minor note: we should be concerned about division by zero, but consider that the problem is symmetric, so if one of the two given angles is zero we can, if needed, rename that angle alpha and call the other (non-zero) angle beta, reversing the roles of A and B as well. This will suffice to guarantee that the ratio above is well defined. An alternative approach to the zero angle problem is given in the algorithm below.)
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| With this substitution the equation becomes
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| :<math>\frac{\sin x}{\sin y}=\tan \phi.</math>
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| We can use two known [[trigonometric identity|trigonometric identities]], namely
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| :<math>\tan \left(\frac{\pi}{4}-\phi\right) = \frac{1- \tan \phi}{\tan \phi +1}</math> and | |
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| :<math>\frac{\tan [(x-y)/2]}{\tan [(x+y)/2]}=\frac{\sin x- \sin y}{\sin x + \sin y}</math>
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| to put this in the form of the second equation we need: | |
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| :<math>\tan \frac{1}{2}(x-y) = \tan \frac{1}{2}(\alpha+\beta+C) \tan \left(\frac{\pi}{4}-\phi\right).</math>
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| We now need to solve these two equations in two unknowns. Once ''x'' and ''y'' are known the various triangles can be solved straightforwardly to determine the position of P.<ref>Bowser: A treatise</ref> The detailed procedure is shown below.
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| ==Solution algorithm==
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| We are given two lengths ''AC'' and ''BC'', and three angles <math>\alpha</math>, <math>\beta</math> and ''C''. The solution proceeds as follows:
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| *calculate <math>\phi= \operatorname{atan2}( \rm{BC} \sin \alpha, \rm{AC} \sin\beta )</math>. Where [[atan2]] is a computer function, also called the arctangent of two arguments, that returns the arctangent of the ratio of the two values given. Note that in [[Microsoft Excel]] the two arguments are reversed, so the proper syntax in Excel would be '=atan2(AC*sin(beta), BC*sin(alpha))'. The atan2 function correctly handles the case where one of the two arguments is zero.
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| *calculate <math>K = 2 \pi -\alpha-\beta-C.</math>
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| *calculate <math>W = 2 *\operatorname{atan}[ \tan(\pi/4 - \phi) \tan(\frac{1}{2}(\alpha+\beta+C))].</math>
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| *find <math>x = (K+W)/2</math> and <math>y = (K-W)/2.</math>
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| *if <math>|\beta|>|\alpha|</math> calculate <math>\rm{PC} = \frac{\rm{BC} \sin y}{\sin \beta}</math> else use <math>\rm{PC} = \frac{\rm{AC} \sin x}{\sin \alpha}.</math>
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| *find <math>\rm{PA} = \operatorname{sqrt}(\rm{AC}^2+\rm{PC}^2 - 2*\rm{AC}*\rm{PC}*cos(\pi-\alpha-x)).</math> (This comes from the [[law of cosines]].)
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| *find <math>\rm{PB} = \operatorname{sqrt}(\rm{BC}^2+\rm{PC}^2 - 2*\rm{BC}*\rm{PC}*\cos(\pi-\beta-y)).</math>
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| If we know the coordinates of ''A'': ''x<sub>A</sub>,y<sub>A</sub>'' and ''C'': ''x<sub>C</sub>,y<sub>C</sub>'' in some appropriate coordinate system then we can find the coordinates of ''P'' as well.
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| ==Geometric (graphical) solution==
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| By the [[inscribed angle theorem]] the locus of points from which AC subtends an angle <math>\alpha</math> is a circle having its center on the midline of AC; from the center O of this circle AC subtends an angle <math>2 \alpha</math>. Similarly the locus of points from which CB subtends an angle <math>\beta</math> is another circle. The desired point P is at the intersection of these two loci.
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| Therefore on a map or nautical chart showing the points A, B, C, the following graphical construction can be used:
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| *Draw the segment AC, the midpoint M and the midline, which crosses AC perpendicularly at M. On this line find the point O such that <math>MO=\frac{AC}{2 \tan \alpha}</math>. Draw the circle with center at O passing through A and C.
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| *Repeat the same construction with points B, C and the angle <math>\beta</math>.
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| *Mark P at the intersection of the two circles (the two circles intersect at two points; one intersection point is C and the other is the desired point P.)
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| This method of solution is sometimes called '''Cassini's method'''.
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| ==Rational trigonometry approach==
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| The following solution is based upon a paper by N. J. Wildberger.<ref>[http://www.math.sc.chula.ac.th/cjm/node/46 Greek Geometry, Rational Trigonometry, and the Snellius – Pothenot Surveying Problem]</ref> It has the advantage that it is almost purely algebraic. The only place trigonometry is used is in converting the [[angles]] to [[Rational_trigonometry#Spread|spreads]]. There is only one [[square root]] required.
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| *define the following:
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| :<math>s(x) = \sin^2(x)</math>
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| :<math>A(x,y,z) = (x + y + z)^2 - 2(x^2 + y^2 + z^2)</math>
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| :<math>r_1 = s(\beta)</math>
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| :<math>r_2 = s(\alpha)</math>
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| :<math>r_3 = s(\alpha + \beta)</math>
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| :<math>Q_1 = BC^2</math>
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| :<math>Q_2 = AC^2</math>
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| :<math>Q_3 = AB^2</math>
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| *now let:
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| :<math>R_1 = r_2 Q_3 / r_3</math>
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| :<math>R_2 = r_1 Q_3 / r_3</math>
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| :<math>C_0 = ((Q_1 + Q_2 + Q_3) (r_1 + r_2 + r_3) - 2 (Q_1 r_1 + Q_2 r_2 + Q_3 r_3))/(2 r_3)</math>
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| :<math>D_0 = r_1 r_2 A(Q_1,Q_2,Q_3)/r_3</math>
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| *the following equation gives two possible values for <math>R_3</math>:
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| :<math>(R_3 - C_0)^2 = D_0</math>
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| *choosing the larger of these values, let:
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| :<math>v_1 = 1 - (R_1 + R_3 - Q_2)^2 / (4 R_1 R_3)</math>
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| :<math>v_2 = 1 - (R_2 + R_3 - Q_1)^2 / (4 R_2 R_3)</math>
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| *finally we get:
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| :<math>AP^2 = v_1 R_1 / r_2 = v_1 Q_3 / r_3</math>
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| :<math>BP^2 = v2 R_2 / r_1 = v_2 Q_3 / r_3</math>
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| ==The indeterminate case==
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| When the point P happens to be located on the same circle as A, B and C, the problem has an infinite number of solutions; the reason is that from any other point P' located on the arc APB of this circle the observer sees the same angles alpha and beta as from P ([[inscribed angle theorem]]). Thus the solution in this case is not uniquely determined.
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| The circle through ABC is known as the "danger circle", and observations made on (or very close to) this circle should be avoided. It is helpful to plot this circle on a map before making the observations.
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| A theorem on [[cyclic quadrilateral]]s is helpful in detecting the indeterminate situation. The quadrilateral APBC is cyclic iff a pair of opposite angles (such as the angle at P and the angle at C) are supplementary i.e. iff <math>\alpha+\beta+C = k \pi, (k=1,2,\cdots)</math>. If this condition is observed the calculations should be stopped and an error message ("indeterminate case") returned.
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| ==Solved examples==
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| (Adapted form Bowser,<ref>Bowser: A treatise</ref> exercise 140, page 203). A, B and C are three objects such that ''AC'' = 435 ([[yard]]s), ''CB'' = 320, and ''C'' = 255.8 degrees. From a station P it is observed that ''APC'' = 30 degrees and ''CPB'' = 15 degrees. Find the distances of ''P'' from ''A'', ''B'' and ''C''. (Note that in this case the points C and P are on the same side of the line AB, a different configuration from the one shown in the figure).
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| Answer: ''PA'' = 790, ''PB'' = 777, ''PC'' = 502.
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| A slightly more challenging test case for a computer program uses the same data but this time with ''CPB'' = 0. The program should return the answers 843, 1157 and 837.
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| ==Naming controversy==
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| [[File:Plaquete huis Willebrord Snellius.jpg|thumb|right|300px|Plaque on Snellius' house in Leiden]]
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| The British authority on geodesy, [[George Tyrrell McCaw]] (1870–1942) wrote that the proper term in English was '''Snellius problem''', while '''Snellius-Pothenot''' was the continental European usage.<ref>{{cite journal |first=G. T. |last=McCaw |title=Resection in Survey |journal=The Geographical Journal |volume=52 |issue=2 |year=1918 |pages=105–126 |doi= |jstor=1779558 }}</ref>
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| McCaw thought the name of [[Laurent Pothenot]] (1650–1732) did not deserve to be included as he had made no original contribution, but merely restated Snellius 75 years later.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *Edward A. Bowser: ''A treatise on plane and spherical trigonometry'', Washington D.C., Heath & Co., 1892, page 188 [http://books.google.com/books?id=9MlHAAAAIAAJ Google books]
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| {{DEFAULTSORT:Snellius-Pothenot problem}}
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| [[Category:Trigonometry]]
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| [[Category:Surveying]]
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| [[Category:Mathematical problems]]
| |
Surely the second option would be more beneficial for any website. It is very easy to customize plugins according to the needs of a particular business. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed.
Creating a website from scratch can be such a pain. Should you loved this short article and you wish to receive more info relating to wordpress backup i implore you to visit the web page. If a newbie missed a certain part of the video then they could always rewind. A Wordpress plugin is a software that you can install into your Wordpress site. Now, I want to anxiety that not every single query will be answered. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available.
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The disadvantage is it requires a considerable amount of time to set every thing up. I didn't straight consider near it solon than one distance, I got the Popup Ascendancy plugin and it's up and lengthways, likely you make seen it today when you visited our blog, and I yet customize it to fit our Thesis Wound which gives it a rattling uncomparable visage and search than any different popup you know seen before on any added journal, I hump arrogated asset of one of it's quatern themes to make our own. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. The most important plugins you will need are All-in-One SEO Pack, some social bookmarking plugin, a Feedburner plugin and an RSS sign up button. Where from they are coming, which types of posts are getting top traffic and many more.
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