ࡱ> g! Oh+'0  0 < H T`hpxIntroductory Remarks onntrZIDIDNormalcMiljenko Lapainerks22jMicrosoft Word 8.0s@ZD3@ ɱ @: @_U ljenko\WP\RADOVI\PRAVAC\pravac2.doc=<@ FMicrosoft Equation 3.0 DS Equation Equation.39q  TU-Graz%   Introductory Remarks onIntroductory Remarks on TitleTitel (RZ _PID_GUID _PID_HLINKSAN{648A9D82-68DE-11D2 Straight Line Equation Miljenko Lapaine University of Zagreb Abstract The paper shows that the general solution of the system of differentianl equations of the straight line in polar form can be obtained in several ways. 1. Introduction A curve in a plane Cartesian coordinate system can be given by the equations in parametric form: (1.1) The equations (1.1) determine a straight line in the plane x,y if the functions x=x(s), y=y(s) are a solution of the system of differential equations  (1.2) The system (1.2) can be solved by the method of direct integration:  (1.3) and then  (1.4) A1, A2, B1 and B2 are constants in the equations (1.4). Let us notice that the parameter s can be any parameter. However, e.g. if we put the condition that s is a natural parmeter, i.e. thae arc lenght, this means  (1.5) or  (1.6) In the book Geometry, Relativity, Geodesy (Moritz, Hofmann-Wellenhof 1993) its authors derive the straight line equation in general curvilinear coordinates in the plane and as an example they give the following system of second order differential equations:  (1.7) They say: "...which is the straight line in polar coordinates. It is not so easy to see that these two differential equations of second order for r and for  represent a straight line. However, it is fairly simple to see that differential equations are satisfied by the equation of a straight line as shown in Fig.":  (1.8) Fig. 1 Moritz and Hofmann-Wellenhof compute the first and second derivatives of the functions r=r(s) and =(s) given by (1.8) and after substituting in (1.7), they conclude that (1.8) is realy a solution of the system (1.7). In the following chapter it will be shown that the system (1.7) is not so hard to solve, as it can be concluded from the previous quotation. It will be enough to transform the problem from polar to Cartesian coordinate system. Besides, assuming that s is a natural parameter, it will be shown in two ways, that the system (1.7) can be solved without the transformation to Cartesian coordinate system. 2. Solution by transformation to the Cartesian coordinate system The result of this chapter will be formulated in the following form: Theorem 1. The general solution of the system (1.7) is:  (2.1) If s is a natural parameter, then the general solution of (1.7) is:  (2.2) If in addition there are initial conditions r(0)=p, (0)=, r'(0)=0, then it is  (1.8) In the formulas (2.1), (2.2) and (1.8) A1, A2, B1, B2, C1, C2, C3,  and p are constants. Proof: The formulas connecting polar and Cartesian coordinate systems are well known:  (2.3) By derivation of (2.3) we obtain  (2.4)  and the  (2.5)  from where, taking into account (1.7), it follows immediatelly  (1.2) Further, as it was explained in the introductory chapter (1.4) If we want to write the solution in polar coordinate system, we may substitute (1.4) into (2.3)  (2.6) and then by squaring and adding or dividing, the following expresions can be derived:  (2.1) If s is a natural parameter, then (1.6) is valid, and the substitution  (2.7)  gives  (2.2) which can be easily checked. Finally, if the initial conditions are given: r(0)=p, (0)=, r'(0)=0, then from (2.2) immediatelly follows  (2.8) The relations (2.2) and (2.8) give  (1.8) which proofs the theorem 1. 3. Solution in polar coordinate system In this chapter it will be proved, without using the transformation to Cartesian coordinate system, that the general solution of the system of differential equations (1.7) is given by (2.2) providing that s is a natural parameter. That will be shown in two ways. Proof 1. The first differential equation from the system (1.7) is:  (3.1) The first fundamental form can be obtained if (2.4) is substituted into (1.5):  (3.2) If the equation (3.1) is multiplied by r and added to (3.2) the result will be  (3.3) Let us notice that (3.3) can be writen in the form  (3.4) From (3.4) it follows  (3.5)  (3.6) and after integrating  (3.7) which is the first set of functions in (2.2). If (3.7) is substituted in the first fundamental form (3.2) we will obtain  (3.8)  from where after integrating  (3.9) and this is the second set of functions in (2.2). Let us notice that in this proof we did not use the second equation of the system (1.7). It is not hard to check that the obtained solution (3.7) and (3.9) also fullfils the second equation of the system (1.7). This ends the proof. Proof 2. Let us take the second equation from the system (1.7):  (3.10) and the first fundamental form  (3.2) From the last relation it can be written up to the sign  (3.11) and than to compute  (3.12) If the first and second derivative of  is substituting from (3.11) and (3.12) into (3.10), after the appropriate trasnformation we can get  (3.13) From (3.13) we conclude that it should be  (3.14) or  (3.15) If (3.14) is substituting into (3.1) or into the first of equations (1.7) it follows  (3.16) which is impossible because of (3.2) or because of the first fundamental form. This means that the solution should be found on the bases of the equation (3.15). But (3.15) is equivalent to (3.3), which means that further reasoning is identical to the one former described. References Moritz, H., Hofmann-Wellenhof, B. (1993): Geometry, Relativity, Geodesy. Wichmann Verlag, Karlsruhe.  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