Wolfe net

Wolfe equatorial grid with a coordinate step of 10 °

The Wolfe grid in crystallography is a stereographic equatorial projection of the degree grid of the sphere from the center of the projection located at its equator , carried out on the plane of the Meridian , which is 90 ° distant from the selected center. This meridian is called the main meridian of the grid. The meridians and parallels of the Wolfe grid play an auxiliary role as projections of the arcs of large and small circles of the sphere. The points of convergence of the meridians are called grid poles; the line segment connecting the grid poles is called the grid axis; a straight line segment equidistant from the poles and perpendicular to the axis is called the equator of the grid.

All constructions and transformations using the Wolfe grid are carried out on tracing paper on which the center of the grid, its main meridian, axis and equator are transferred, as well as points, whose spherical coordinates are to be converted, are plotted. Turnings of tracing paper are made with preservation of centering relative to the grid.

The Wolfe grid is usually built with a coordinate step of 2 °.

The method was invented by crystallographer George Wolfe .

Application Examples

The Wolfe grid allows you to graphically, without additional calculations, solve many problems of geometric crystallography related to the angular characteristics of crystals, as well as navigation and astrometric problems.

Using the Wolfe grid, a stereographic equatorial projection of a point, given by its spherical coordinates, is built. $\phi$ ${\ displaystyle \ phi}$ ₁ and $\rho$ ${\ displaystyle \ rho}$ ₁ . By rotating the tracing paper around the center of the grid at the required angle, taking into account its sign, the resulting coordinates of the point are obtained $\phi$ ${\ displaystyle \ phi}$ ₂ and $\rho$ ${\ displaystyle \ rho}$ ₂ on the grid. Depending on the class of tasks to be solved, the coordinates of points on the grid can be specified in various ways.

In crystallography, the following order of indication of coordinates is adopted: angles $0^{\circ }\leq \phi <360^{\circ }$ ${\ displaystyle 0 ^ {\ circ} \ leq \ phi <360 ^ {\ \ circ}}$ are counted along the circumference of the Wulff grid, a positive direction clockwise, starting from the right end of its equator; corners $0^{\circ }\leq \rho \leq 180^{\circ }$ ${\ displaystyle 0 ^ {\ circ} \ leq \ rho \ leq 180 ^ {\ \ circ}}$ - along the axis and equator, from the center of the grid, while the range $90^{\circ }<\rho \leq 180^{\circ }$ ${\ displaystyle 90 ^ {\ circ} <\ rho \ leq 180 ^ {{\ circ}}$ corresponds to the projections of points lying under the plane of the main meridian. The center of the grid corresponds to the coordinates $\rho =0^{\circ }$ ${\ displaystyle \ rho = 0 ^ {\ \ circ}}$ and $\rho =180^{\circ }$ ${\ displaystyle \ rho = 180 ^ {{\ circ}}$ ; right end of the equator - $\rho =90^{\circ },\phi =0^{\circ }$ ${\ displaystyle \ rho = 90 ^ {{\ circ}, \ phi = 0 ^ {\ circ}}$ ; the left end of the equator - $\rho =90^{\circ },\phi =180^{\circ }$ ${\ displaystyle \ rho = 90 ^ {\ \ circ}, \ phi = 180 ^ {\ \}}}$ ; "upper" pole - $\rho =90^{\circ },\phi =270^{\circ }$ ${\ displaystyle \ rho = 90 ^ {\ \ circ}, \ phi = 270 ^ {\ \}}}$ ; "lower" pole - $\rho =90^{\circ },\phi =90^{\circ }$ ${\ displaystyle \ rho = 90 ^ {{\ circ}, \ phi = 90 ^ {\ \}}}$ .

In the geodetic, navigation or astrographic application of the grid, the following coordinate order is adopted: angles $-90^{\circ }\leq \phi \leq +90^{\circ }$ ${\ displaystyle -90 ^ {\ circ} \ leq \ phi \ leq +90 ^ {\ circ}}$ corresponding to the latitude, declination or height above the horizon, are measured around the wolfe grid, the positive direction is clockwise, starting from the left end of its equator; corners $0^{\circ }\leq \lambda <360^{\circ }$ ${\ displaystyle 0 ^ {\ circ} \ leq \ lambda <360 ^ {\ \ circ}}$ corresponding to longitude, right ascension or hour angle - along the equator of the grid from its right end. Positions of points with coordinates $\lambda >180^{\circ }$ ${\ displaystyle \ lambda> 180 ^ {\ circ}}$ are according to the rule $360^{\circ }-\lambda$ ${\ displaystyle 360 ^ {\ circ} - \ lambda}$ . The center of the grid has coordinates $\lambda =90^{\circ }$ ${\ displaystyle \ lambda = 90 ^ {\ circ}}$ and $\lambda =270^{\circ }$ ${\ displaystyle \ lambda = 270 ^ {\ circ}}$ .

In the context of solving navigation problems, the grid can represent the required system of spherical coordinates, for example, the equatorial , then the north pole is mapped to the upper pole of the grid, the south pole to the lower pole of the grid, the celestial equator to the equator of the grid; the meridian of the observer coincides with the main meridian of the grid. Zenith and nadir are located at points corresponding to the latitude of the observer’s location: at points $(\phi ,\lambda =180^{\circ })$ ${\ displaystyle (\ phi, \ lambda = 180 ^ {\ circ})}$ and $(-\phi ,\lambda =0^{\circ })$ ${\ displaystyle (- \ phi, \ lambda = 0 ^ {\ \}})}$ respectively. In this case, the declination of the luminaries is measured along the main meridian, and hour angles are measured along the equator of the grid.

When using a horizontal coordinate system - zenith and nadir are in the corresponding poles of the grid, the equator of the grid corresponds to the true horizon of the observer. The observer meridian coincides with the main meridian of the grid. The poles of the world are on the main meridian in points $(90^{\circ }-\phi )$ ${\ displaystyle (90 ^ {\ circ} - \ phi)}$ and $(-90^{\circ }+\phi )$ ${\ displaystyle (-90 ^ {\ circ} + \ phi)}$ respectively. The point of north (N) is displayed on the right end of the equator, the point of the south (S) - on the left, the points of the east and west - in the center of the grid. In this case, the heights of the luminaries above the horizon are measured along the main meridian of the grid (from the point of the south); along the equator of the grid (from the point of the north) are true bearings of the luminaries.

By rotating the tracing paper around the center of the grid at the appropriate angle, the coordinates of the sun are converted from horizontal to equatorial coordinate system and back.

Wolfe Mesh Method

Wolfe Mesh Order

Let us use the property of the stereographic equatorial projection that the meridians and parallels of the Wulff grid are arcs of circles.

Draw a circle of radius $R$ ${\ displaystyle R}$ centered on $C$ ${\ displaystyle C}$ , build two mutually perpendicular diameters $P_{1}-P_{2}$ ${\ displaystyle P_ {1} -P_ {2}}$ and $Q_{1}-Q_{2}$ ${\ displaystyle Q_ {1} -Q_ {2}}$ . Positive angle values $\phi$ ${\ displaystyle \ phi}$ counted clockwise from point $Q_{2}$ ${\ displaystyle Q_ {2}}$ . Selecting the desired grid step - find on the circle $P_{1}Q_{2}P_{2}Q_{1}$ ${\ displaystyle P_ {1} Q_ {2} P_ {2} Q_ {1}}$ auxiliary point $A_{\phi }$ ${\ displaystyle A _ {\ phi}}$ measuring an arc on a circle $Q_{2}-A_{\phi }$ ${\ displaystyle Q_ {2} -A _ {\ phi}}$ multiple of the selected step angle $\phi$ ${\ displaystyle \ phi}$ . Find on the beam $C-P_{2}$ ${\ displaystyle C-P_ {2}}$ auxiliary point $O_{\phi }$ ${\ displaystyle O _ {\ phi}}$ lying at a distance $r_{\phi }={\frac {R}{\tan \phi }}$ ${\ displaystyle r _ {\ phi} = {\ frac {R} {\ tan \ phi}}}$ from point $A_{\phi }$ ${\ displaystyle A _ {\ phi}}$ . Taking point $O_{\phi }$ ${\ displaystyle O _ {\ phi}}$ as a center, draw from a point $A_{\phi }$ ${\ displaystyle A _ {\ phi}}$ arc of radius $r_{\phi }$ ${\ displaystyle r _ {\ phi}}$ inside the circle; parallel latitude $\phi$ ${\ displaystyle \ phi}$ built. The parallels of the second half of the grid are constructed in the same way, but the corners $\phi$ ${\ displaystyle \ phi}$ counted from the point $Q_{1}$ ${\ displaystyle Q_ {1}}$ and auxiliary points are located on the ray $C-P_{1}$ ${\ displaystyle C-P_ {1}}$ .

To build the meridians of the grid with the selected step, calculate the position of the auxiliary point $O_{\lambda }$ ${\ displaystyle O _ {\ lambda}}$ on the beam $C-Q_{2}$ ${\ displaystyle C-Q_ {2}}$ on distance $r_{\lambda }={\frac {R}{\sin \lambda }}$ ${\ displaystyle r _ {\ lambda} = {\ frac {R} {\ sin \ lambda}}}$ from any pole. Taking point $O_{\lambda }$ ${\ displaystyle O _ {\ lambda}}$ as a center, draw between the poles $P_{1}$ ${\ displaystyle P_ {1}}$ and $P_{2}$ ${\ displaystyle P_ {2}}$ arc of radius $r_{\lambda }$ ${\ displaystyle r _ {\ lambda}}$ ; longitude meridian $\lambda$ ${\ displaystyle \ lambda}$ built by The meridians of the second half of the grid are constructed in the same way, but the auxiliary points are located on the ray $C-Q_{1}$ ${\ displaystyle C-Q_ {1}}$ .

Links

Wolfe Grid - an article from the Great Soviet Encyclopedia .
Wolf G.V. A way to graphically solve problems in cosmography and mathematical geography. - Nizhny Novgorod, 1909.

Wolfe net

Application Examples

Wolfe Mesh Method

Links

More articles: