![\"\"](\"https:\/\/www.lunarip.com\/wp-content\/uploads\/2021\/03\/polynomials.png\")
<\/figure>\n\n\n\nin these examples, the single x value is mapped onto the output y value for the range[-5,5] of x. <\/p>\n\n\n\n
Space filling curve<\/h2>\n\n\n\n
A space filling curve is a parameterized function which maps a unit line segment to a continuous curve in the unit space (square, cube, ….). For the unit square form we have i in range[0..1] with a function F(i) that provides output x,y with 0<=x,y,<=1.<\/p>\n\n\n\n
The Hilbert Curve<\/h3>\n\n\n\n
The Hilbert curve<\/a> is a curve which uses recursion in its definition and it is a continuous fractal so as its order increases it gets more dense but maintains the basic properties of the curve and the space that the curve fills. <\/p>\n\n\n\n In the space in which this curve is inscribed (x,y) where 0<=x,y,<=1 we can see that as the order (complexity) of the curve increases, it does a better job of filling in the space. Another way to consider this is that the x and y input for low order are like low resolution numbers along the lines of the resolution of an Analog to Digital (A\/D) convertor<\/a>. <\/p>\n\n\n\n A 1 bit A\/D convertor can only indicate the presence or absence of a signal while a 12bit A\/D convertor can capture significant signal resolution. <\/p>\n\n\n\n The lowest order Hilbert curve starts with x and y values being either 0 or 1 which gives us the simple “U” pattern which have x,y coordinates of [(x=0,y=1),(0,0),(1,0),(1,1)] <\/p>\n\n\n\n At each point along the curve it also has a value in the range [0,1], so again the simple example, would have the function map<\/p>\n\n\n\n (0,1) -> 0, (0,0)->0.33, (1,0)-> 0.66 and (1,1)->1<\/p>\n\n\n\n As the x,y resolution of the curve increases then its possible for number of curve values to become larger to the point where is we leave the A\/D convertor example where x and y become real numbers then the curve values can also be a real number between 0 and 1. <\/p>\n\n\n\n If we say that our input parameters are (x,y) and curve output value is z at point (x,y) then the hilbert curve has an interesting and very attractive property that values of x and y representing points that are close to each other will generate values of z that are also close to each other. This property allows for (in our example) 2 dimensional closeness to be mapped onto numbers that retain the ability to measure closeness but using only 1 dimension. <\/p>\n\n\n\n So far I have been using a 2 dimensional example where the space filling curve traverses a 2 dimensional space, but its also possible to have a 3 dimensional curve that traverse a 3 dimensional space and also in higher dimensions. This effectively provides a method for high dimensional data to be mapped onto a simple dimensional value where some inherent properties of the high dimension data are maintained in the linear representation. <\/p>\n\n\n\n![\"\"](\"https:\/\/www.lunarip.com\/wp-content\/uploads\/2021\/03\/Hilbert-curve_rounded-gradient-animated.gif\")