Distance fields for high school geometry Mark Sawula Peddie School 17 Apr 10 (work in progress)

Outline Context How I became interested What is a scalar field? What is a distance field? Distance fields: Points Distance fields: Lines Distance fields: Points & Lines Closing

Context Geometry Honors Pedagogy Problem-based (Exeter-style) Student-centered Content Euclidean geometry Analytic geometry Transformational geometry “Other Goals” Develop mathematical maturity Develop problem- solving capabilities Articulation Develop algebra skills Bridge to Algebra II topics (eg, locus) Lay foundation for Precalculus & Calculus topics Technology Laptops Geogebra Wiki (as website)

How I became interested in Distance Fields (2006)

What is a scalar field? A ‘(s)calar field associates a scalar value to every point in a space.’ ( One example of a scalar field is a topographic map. The elevation is assigned to every point on the map. Contour lines connect points at the same elevation. You can ask the usual good questions related to the spacing of contour lines.

What is a scalar field? (2) A slope field is actually an example of a vector field rather than a scalar field. Each point is associated with a direction and a magnitude rather than just a scalar. Nonetheless, working with scalar fields should lay groundwork for later work with slope fields. gif/SlopeField_700.gif

What is a distance field? A distance field is a scalar field in which the scalar is the distance to an object or set of objects (or points). Distance fields are used in image processing.

What is a distance field? (2) From Wikipedia ( ) : A distance transform, also known as distance map or distance field, is a derived representation of a digital image. The choice of the term depends on the point of view on the object in question: whether the initial image is transformed into another representation, or it is simply endowed with an additional map or field.digital imagepoint of view The map labels each pixel of the image with the distance to the nearest obstacle pixel. A most common type of obstacle pixel is a boundary pixel in a binary image. See the image for an example of a chessboard distance transform on a binary image.pixelbinary imagechessboard distancebinary image Usually the transform/map is qualified with the chosen metric. For example, one may speak of Manhattan distance transform, if the underlying metric is Manhattan distance. Common metrics are:metricManhattan distance Euclidean distance Taxicab geometry, also known as City block distance orManhattan distance. Taxicab geometry Chessboard distance Applications are digital image processing (e.g., blurring effects,skeletonizing), motion planning in robotics, and even pathfinding.digital image processingskeletonizingmotion planningroboticspathfinding

Distance fields: Points (1) The picture to the right shows a ‘distance field’, regions in which the distance to something red is indicated by a shade of green. Locations that are very far from anything red are shaded dark green; locations that are very close to something red are nearly white. The figure to the right has a single red dot in the center. Every other location is a different shade of green based upon how far away it is from the center dot. All of the locations that are the same distance from the center dot will be the same color. What shape will all the locations that are the same color make? Why?

Distance fields: Points (2) Suppose the coordinates of the red point are (2,-3). What is the equation of the contour line for all points a distance of 5 away? What is true of all of the points inside the orange circle? What is true of all of the points outside the orange circle?

Distance fields: Points (3) 5. The figure to the right shows the previous distance field with a few changes. Suppose that C=(5,8) and that N=(t+4,t). Find the length of NC. 6. (continuation) Consider your answer to #5 as a function which takes a value of t as an input and outputs the length of NC. Produce a graph of this function. You may use Geogebra, a graphing calculator, or just pencil and paper. What sort of shape is this? 7. (continuation) Using your answer to #6, find the point on line AB that is closest to point C. 8. (continuation) Without using coordinate geometry, how would you describe the location of the point on line AB closest to C? Does your answer to #7 satisfy your description? Why should this be true?

Distance fields: Points (4) 5. The figure to the right shows the previous distance field with a few changes. Suppose that C=(5,8) and that N=(t+4,t). Find the length of NC.

Distance fields: Points (5)

Distance fields: Points (6)

Distance fields: Points (7) You can discover using analytic geometry or Euclidean geometry that the shortest distance is along a perpendicular to the line through the point. Distance fields give you a way to connect that distance with the radius of a tangent circle, foreshadowing a circle theorem that typically occurs later in the curriculum.

Distance fields: Points (8) Now consider the case of a distance field where the objects are two points. What are the different shapes of possible contour lines?

Distance fields: Points (9) The image above was created using Maple. Thanks, Tim! Under what circumstances do you get each shape of contour? Explore regions of the plane created by contour lines? Relate contour of both points to contours of each point using Venn diagram-like reasoning.

Distance fields: Points (10) What’s up with this line down the middle? Is it a contour? Suppose the two points are (2,7) and (-8,3). What is the “line down the middle of the picture”?

Distance fields: Points (11) You can also import the problem to GeoGebra or Sketchpad (or use paper!) and check to see that the “ghost line” is the perpendicular bisector of the segment connecting the two points.

Distance fields: Points (12)

Distance fields: Points (13) In the distance field graph for 3 points, depicted at left, what shape are the contour lines? the boundaries? What is true of the regions determined by the boundaries? What is true of the intersection point of the boundaries?

Distance fields: Points (14) What shape are the contour lines? Where are the boundaries? What is true of the regions determined by the boundaries? What is true of the intersection point of the boundaries?

Distance fields: Points (15) The Maple version of the image.

Distance fields: Points (16) For the points D = (-2,3), E = (3,4), and F = (2,-1), find a point that is equidistant to all three. Do this first using algebra and coordinate geometry, then confirm with Geogebra. How can you be sure that your answer was correct?

Distance fields: Points (17) When a distance field has several 'red' points, you get a structure like the ones below. Notice that each of the boundaries is made up of points equidistant to two red points. For each boundary, you should be able to identify which red points. Not only are they the closest ones to the boundary, but the boundary is also the perpendicular bisector of the segment between the two points.

Distance fields: Points (18) /20/Coloured_Voronoi_2D.svg/220px- Coloured_Voronoi_2D.svg.png If you were to draw only the 'red' points and the boundaries, you would have what is called a Voronoi diagram, named after the Ukrainian mathematician, Georgy Voronoy ( ). Applications of Voronoi diagrams can be found in astronomy, chemistry, biology, and computer science.

Distance fields: Points (19) See if you can beat the various computer opponents. Which is the easiest? Which is the hardest? How can you make a move that changes only one polygon? Give an example. How can you make a move that changes more than one polygon? Give an example. Go to and play the Voronoi game! Pull down the ‘File ‘ tab and switch to the “Info” tab.

Distance fields: Points (20) Let’s Play!

Distance fields: Points (21) Another useful applet is VoroGlide: hagen.de/GeomLab/VoroGl ide/index.html.en hagen.de/GeomLab/VoroGl ide/index.html.en how many boundaries can meet at one point? how do you make that happen? what kinds of quadrilaterals have boundaries that meet at one point? hagen.de/GeomLab/VoroGlide/inde x.html.en

Distance fields: Points (22) Sketch the Voronoi diagram for the four points above.

Distance fields: Points (23) Here are the points with the perpendicular bisectors drawn.

Distance fields: Points (24) Alternate way of looking at concurrence of perpendicular bisectors. Why can’t this happen?

Distance fields: Lines (1) What is a distance field for a straight line? What do the contour lines look like?

Distance fields: Lines (2) What is a distance field for a two straight lines? What do the contour lines look like? What are those “ghost lines”?

Distance fields: Lines (3) What is a distance field for a triangle? What do the contour lines inside the triangle look like? What do the contour lines outside the triangle look like? What are those “ghost lines”?

Distance fields: Point & Line “I will never look at parabolas the same way again.” --student

Next steps Write this up! Classify quadrilaterals Any other ideas?