Markvorsen, Steen

We consider a specific function of two variables whose graph surface resembles a blue lagoon. The function has a saddle point $p$, but when the function is restricted to any given straight line through $p$ it has a {\em{strict local minimum}} along that line at $p$

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A Blue Lagoon FunctionA BLUE LAGOON FUNCTIONPRINTED IN 3D AT DTU MATHEMATICS UPON SPECIAL REQUESTFROM GUNNAR MOHR, DEAN OF STUDIES, DECEMBER 2007Abstract. We consider a specific function of two variables whosegraph surface resembles a blue lagoon. The function has a saddlepoint p, but when the function is restricted to any given straightline through p it has a strict local minimum along that line at p.1. Definition and propertiesA function f(u, v) is defined in R2 as follows:f(u, v) =(1− (u− 1)2 − v2) (4− (u− 2)2 − v2) .The function is zero along the two circles (the red circles in Figure 1):(u− 1)2 + v2 = 1 and (u− 2)2 + v2 = 4 .The point of interest is p, where the two red circles meet. This pointhas coordinates p = (0, 0). It is a stationary point for f :∇f |(0,0) = 0 .The Hessian of f is positive semi-definite at p :Hess f |(0,0) =(2 00 0).In the disc domain shown in Figure 1 there are two subdomains, wherethe function is positive (the green subdomains), and one subdomainwhere the function is negative (the blue subdomain). Every straightline through p therefore only experiences positive values of f close top - except precisely at p, where the value is 0. The point p is thencea strict local minimum along every one of these straight lines. Theyellow circle marks the location of the local maxima along the respec-tive straight lines through p. The function (considered as a functionin R2) does not itself have a local minimum at p . For example, thefunction is decreasing from p along the blue circle through p in theblue subdomain in between the two red circles through p in Figure 1(see the precise analysis on page 5). The point p is thus a saddle pointin the sense that it is a stationary point with the property that everyneighborhood around p contains points where f is strictly larger thanf(p) = 0 as well as points where f is strictly smaller than 0 .2000 Mathematics Subject Classification. Primary 26.Key words and phrases. Functions of two variables.12 A BLUE LAGOON FUNCTION2. Figuresvu Figure 1. Straight lines through the stationary pointand descriptive circles in the considered domain for thefunction f .vfuFigure 2. The graph surface of f looks roughly like theBlue Lagoon in Iceland, see the picture on page 4.A BLUE LAGOON FUNCTION 3vu Figure 3. The function −f unfolded with ’dual colors’.uv-fFigure 4. The graph surface of −f (with ’dual’ col-ors) looks roughly like the Blue Lagoon at Abereiddy inWales, UK.4 A BLUE LAGOON FUNCTIONFigure 5. The Blue Lagoon in Iceland.Figure 6. The Blue Lagoon at Abereiddy in Wales, UK.3. AnalysisAny straight line through p = (0, 0) may be parametrized as followsLw : r(t) = ( t cos(w), t sin(w) ) for t ∈ R and w ∈[−pi2,pi2].When restricting f to Lw we get the restricted function:g(t) = f(r(t)) = f( t cos(w), t sin(w) ) = t2(8 cos2(w)−6t cos(w)+t2) .These restricted functions are displayed in Figure 7 for a couple ofw−values. It is clear from this inspection, that at least for w 6= ±pi/2 ,A BLUE LAGOON FUNCTION 5w = Pi/3 w=Pi/5 w=Pi/12t0 1 2g tK112Figure 7. The line-restricted functions g(t).every g has a local minimum along Lw at p corresponding to t = 0 .This also follows precisely from the derivatives of g at t = 0 :(3.1) g′(0) = 0 and g′′(0) = 16 cos2(w) > 0 .For the special values w = ±pi/2 (corresponding to Lw being thev−axis), we get g(t) = t4 . This shows that the restriction of f to thev−axis also has a strict local minimum at p.The yellow circle in Figure 1 appears as the locus of local max-ima (on the green ’island’) of g along the straight lines Lw for w ∈[−pi/2, pi/2 ]. The blue circle in Figure 1 is correspondingly the locusof local minima (in the blue ’lagoon’) of g along the lines. Indeed,g′(t) = 2t(−9t cos(w) + 2t2 + 8 cos2(w)). Thus g′(t) = 0 for t = 0 ,t = (1/4)(9−√17) cos(w) , and for t = (1/4)(9+√17) cos(w) . Wheninserted into r(t) this gives the point p and the two circles, respectively.In particular we note, that the values of f along the blue circle are,as a function of the direction angle w ∈ [−pi/2, pi/2 ] from the point p :h(w) = − cos4(w)(107 + 51√17)/32 .This function is clearly negative, except at p - corresponding to w =± pi/2 - and it is clearly decreasing when w moves away from thesevalues of w - corresponding to walking (or rather diving) away from palong the blue circle in Figure 2, as claimed on page 1.Department of Mathematics, Technical University of Denmark.

A Blue Lagoon Function

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A Blue Lagoon Function

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