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Two types of cartesian coordinates?
[ For coherent viewing, fixed-width font
(such as "courier new") and "UTF-8" character encoding should be utilized ] On Jun 13, 2:08 am, Nomen Nescio wrote: I have seen two different types of cartesian coordinates: x = a * cos(theta) y = b * sin(theta) and x' = a * sin(theta) y' = b * cos(theta) or x' = b * cos(theta) y' = a * sin(theta) What is the difference between x,y and x',y', in terms of names (identities?) and their basic meanings? I believe x and y are known as the "parametric equation of ellipse", but what about x' and y'? I don't know what the formal name is, but basically x' and y' define the "parametric equation of the ellipse surface" or, extending it to three axes----X', Y', Z'----by adding a longitude, the "parametric equation of the ellipsoid surface". Where ë is the geographical/geodetic longitude; a_x, a_y are the equatorial radii of their respective axis: a(ë) = ((a*cos(ë))^2 + (a*sin(ë))^2)^.5; a_m = b´ = (a_x*a_y)^.5; and b_x = a_x´ = b*(a_y/a_x)^.5 = b*a_y/a_m; b_y = a_y´ = b*(a_x/a_y)^.5 = b*a_x/a_m; b(ë) = a´(ë)=((b_x*cos(ë))^2 + (b_y*sin(ë))^2)^.5; then X = a_x * cos(â) * cos(ë); Y = a_y * cos(â) * sin(ë); x(ë) = (X^2 + Y^2)^.5 = a(ë) * cos(â); y = Z = b * sin(â); R(â) = (x(ë)^2 + y^2)^.5 = (X^2 + Y^2 + Z^2)^.5; and X´ = b_x * cos(â) * cos(ë); Y´ = b_y * cos(â) * sin(ë); x´(ë) = (X´^2 + Y´2)^.5 = b(ë) * cos(â); y´ = Z´ = a_m * sin(â); S(â) = R´(â) = (x´(ë)^2 + y´^2)^.5, = (X´^2 + Y´^2 + Z´^2)^.5; Thus, for an ellipse (and non-scalene spheroid), these reduce to x = a * cos(â); y = b * sin(â); R(â) = S(90-â) = (x^2 + y^2)^.5, = ((a * cos(â))^2 +(b * sin(â))^2)^.5; and x´ = b * cos(â); y´= a * sin(â); S(â) = R(90-â) = (x´^2 + y´^2)^.5, = ((a * sin(â))^2 +(b * cos(â))^2)^.5; So what does this all mean? Well, in terms of the surface parameters, rather than derivatives of â's trig functions, x´ and y´ are fundamentally based on radii complements, as the triaxial case demonstrates. In terms of uses, R(â) is the integrand for the well known elliptic integral of the second kind, and S(â) is the auxiliary integrand for meridional distance, DxM, as well as (authalic) surface area: Where ö is the geographical/geodetic latitude and M is the (conjugate) meridional radius of curvature, M(ö) = (a*b)^2/R(ö)^3, = (a*b)^2/((a * cos(ö))^2 +(b * sin(ö))^2)^1.5; __ â_f __ ö_f / / DxM = / S(â)dâ = / M(ö)dö; __/ __/ â_s ö_s and __ â_f Surface / Area = Äë a / cos(â)*S(â)dâ, __/ â_s __â_f __ ë_f / / = a_m / cos(â) / (x´(ë)^2 + y´^2)^.5 dëdâ __/ __/ â_s ë_s ~Kaimbridge~ ----- Wikipedia--Contributor Home Page: http://en.wikipedia.org/wiki/User:Kaimbridge ***** Void Where Permitted; Limit 0 Per Customer. ***** |
#2
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Two types of cartesian coordinates?
[ For coherent viewing, fixed-width font
(such as "courier new") and "UTF-8" character encoding should be utilized ] On Jun 13, 2:08 am, Nomen Nescio wrote: I have seen two different types of cartesian coordinates: x = a * cos(theta) y = b * sin(theta) and x' = a * sin(theta) y' = b * cos(theta) or x' = b * cos(theta) y' = a * sin(theta) What is the difference between x,y and x',y', in terms of names (identities?) and their basic meanings? I believe x and y are known as the "parametric equation of ellipse", but what about x' and y'? I don't know what the formal name is, but basically x' and y' define the "parametric equation of the ellipse surface" or, extending it to three axes----X', Y', Z'----by adding a longitude, the "parametric equation of the ellipsoid surface". Where ë is the geographical/geodetic longitude; a_x, a_y are the equatorial radii of their respective axis: a(ë) = ((a*cos(ë))^2 + (a*sin(ë))^2)^.5; a_m = b´ = (a_x*a_y)^.5; and b_x = a_x´ = b*(a_y/a_x)^.5 = b*a_y/a_m; b_y = a_y´ = b*(a_x/a_y)^.5 = b*a_x/a_m; b(ë) = a´(ë)=((b_x*cos(ë))^2 + (b_y*sin(ë))^2)^.5; then X = a_x * cos(â) * cos(ë); Y = a_y * cos(â) * sin(ë); x(ë) = (X^2 + Y^2)^.5 = a(ë) * cos(â); y = Z = b * sin(â); R(â) = (x(ë)^2 + y^2)^.5 = (X^2 + Y^2 + Z^2)^.5; and X´ = b_x * cos(â) * cos(ë); Y´ = b_y * cos(â) * sin(ë); x´(ë) = (X´^2 + Y´2)^.5 = b(ë) * cos(â); y´ = Z´ = a_m * sin(â); S(â) = R´(â) = (x´(ë)^2 + y´^2)^.5, = (X´^2 + Y´^2 + Z´^2)^.5; Thus, for an ellipse (and non-scalene spheroid), these reduce to x = a * cos(â); y = b * sin(â); R(â) = S(90-â) = (x^2 + y^2)^.5, = ((a * cos(â))^2 +(b * sin(â))^2)^.5; and x´ = b * cos(â); y´= a * sin(â); S(â) = R(90-â) = (x´^2 + y´^2)^.5, = ((a * sin(â))^2 +(b * cos(â))^2)^.5; So what does this all mean? Well, in terms of the surface parameters, rather than derivatives of â's trig functions, x´ and y´ are fundamentally based on radii complements, as the triaxial case demonstrates. In terms of uses, R(â) is the integrand for the well known elliptic integral of the second kind, and S(â) is the auxiliary integrand for meridional distance, DxM, as well as (authalic) surface area: Where ö is the geographical/geodetic latitude and M is the (conjugate) meridional radius of curvature, M(ö) = (a*b)^2/R(ö)^3, = (a*b)^2/((a * cos(ö))^2 +(b * sin(ö))^2)^1.5; __ â_f __ ö_f / / DxM = / S(â)dâ = / M(ö)dö; __/ __/ â_s ö_s and __ â_f Surface / Area = Äë a / cos(â)*S(â)dâ, __/ â_s __â_f __ ë_f / / = a_m / cos(â) / (x´(ë)^2 + y´^2)^.5 dëdâ __/ __/ â_s ë_s ~Kaimbridge~ ----- Wikipedia--Contributor Home Page: http://en.wikipedia.org/wiki/User:Kaimbridge ***** Void Where Permitted; Limit 0 Per Customer. ***** |
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