A Blue Lagoon Function

Transcription

1 Downloaded from orbit.dt.dk on: Oct 11, 2018 A Ble Lagoon Fnction Markorsen, Steen Pblication date: 2007 Link back to DTU Orbit Citation (APA): Markorsen, S., (2007). A Ble Lagoon Fnction General rights Copyright and moral rights for the pblications made accessible in the pblic portal are retained by the athors and/or other copyright owners and it is a condition of accessing pblications that sers recognise and abide by the legal reqirements associated with these rights. Users may download and print one copy of any pblication from the pblic portal for the prpose of priate stdy or research. Yo may not frther distribte the material or se it for any profit-making actiity or commercial gain Yo may freely distribte the URL identifying the pblication in the pblic portal If yo beliee that this docment breaches copyright please contact s proiding details, and we will remoe access to the work immediately and inestigate yor claim.

2 A BLUE LAGOON FUNCTION PRINTED IN 3D AT DTU MATHEMATICS UPON SPECIAL REQUEST FROM GUNNAR MOHR, DEAN OF STUDIES, DECEMBER 2007 Abstract. We consider a specific fnction of two ariables whose graph srface resembles a ble lagoon. The fnction has a saddle point p, bt when the fnction is restricted to any gien straight line throgh p it has a strict local minimm along that line at p. 1. Definition and properties A fnction f(, ) is defined in R 2 as follows: f(, ) = ( 1 ( 1) 2 2) ( 4 ( 2) 2 2). The fnction is zero along the two circles (the red circles in Figre 1): ( 1) = 1 and ( 2) = 4. The point of interest is p, where the two red circles meet. This point has coordinates p = (0, 0). It is a stationary point for f : f (0,0) = 0. The Hessian of f is positie semi-definite at p : ( ) 2 0 Hess f (0,0) =. 0 0 In the disc domain shown in Figre 1 there are two sbdomains, where the fnction is positie (the green sbdomains), and one sbdomain where the fnction is negatie (the ble sbdomain). Eery straight line throgh p therefore only experiences positie ales of f close to p - except precisely at p, where the ale is 0. The point p is thence a strict local minimm along eery one of these straight lines. The yellow circle marks the location of the local maxima along the respectie straight lines throgh p. The fnction (considered as a fnction in R 2 ) does not itself hae a local minimm at p. For example, the fnction is decreasing from p along the ble circle throgh p in the ble sbdomain in between the two red circles throgh p in Figre 1 (see the precise analysis on page 5). The point p is ths a saddle point in the sense that it is a stationary point with the property that eery neighborhood arond p contains points where f is strictly larger than f(p) = 0 as well as points where f is strictly smaller than Mathematics Sbject Classification. Primary 26. Key words and phrases. Fnctions of two ariables. 1

3 2 A BLUE LAGOON FUNCTION 2. Figres Figre 1. Straight lines throgh the stationary point and descriptie circles in the considered domain for the fnction f. f Figre 2. The graph srface of f looks roghly like the Ble Lagoon in Iceland, see the pictre on page 4.

4 A BLUE LAGOON FUNCTION Figre 3. The fnction f nfolded with dal colors. -f Figre 4. The graph srface of f (with dal colors) looks roghly like the Ble Lagoon at Abereiddy in Wales, UK. 3

5 4 A BLUE LAGOON FUNCTION Figre 5. The Ble Lagoon in Iceland. Figre 6. The Ble Lagoon at Abereiddy in Wales, UK. 3. Analysis Any straight line throgh p = (0, 0) may be parametrized as follows [ L w : r(t) = ( t cos(w), t sin(w) ) for t R and w π 2, π ]. 2 When restricting f to L w we get the restricted fnction: g(t) = f(r(t)) = f( t cos(w), t sin(w) ) = t 2 (8 cos 2 (w) 6t cos(w)+t 2 ). These restricted fnctions are displayed in Figre 7 for a cople of w ales. It is clear from this inspection, that at least for w ±π/2,

6 A BLUE LAGOON FUNCTION 5 2 g t t K1 w = Pi/3 w=pi/5 w=pi/12 Figre 7. The line-restricted fnctions g(t). eery g has a local minimm along L w at p corresponding to t = 0. This also follows precisely from the deriaties of g at t = 0 : (3.1) g (0) = 0 and g (0) = 16 cos 2 (w) > 0. For the special ales w = ± π/2 (corresponding to L w being the axis), we get g(t) = t 4. This shows that the restriction of f to the axis also has a strict local minimm at p. The yellow circle in Figre 1 appears as the locs of local maxima (on the green island ) of g along the straight lines L w for w [ π/2, π/2 ]. The ble circle in Figre 1 is correspondingly the locs of local minima (in the ble lagoon ) of g along the lines. Indeed, g (t) = 2t( 9t cos(w) + 2t cos 2 (w)). Ths g (t) = 0 for t = 0, t = (1/4)(9 17) cos(w), and for t = (1/4)(9+ 17) cos(w). When inserted into r(t) this gies the point p and the two circles, respectiely. In particlar we note, that the ales of f along the ble circle are, as a fnction of the direction angle w [ π/2, π/2 ] from the point p : h(w) = cos 4 (w)( )/32. This fnction is clearly negatie, except at p - corresponding to w = ± π/2 - and it is clearly decreasing when w moes away from these ales of w - corresponding to walking (or rather diing) away from p along the ble circle in Figre 2, as claimed on page 1. Department of Mathematics, Technical Uniersity of Denmark.