Geometry and graphs – transformations, graphs, and technical drawing

What it shows

Three topics share one page. Plane transformations send every point to an image: translations, rotations and reflections preserve distances, while a dilation with ratio k multiplies lengths by |k| and, when k is negative, places the image on the other side of the centre. Graph theory starts from Euler's Seven Bridges of Königsberg: a connected graph has an Eulerian circuit when every vertex has even degree, and an Eulerian path when exactly two vertices have odd degree. Technical drawing shows how a 3D block is represented by its front, top and side views in first-angle projection.

How to use

Choose the Current topic. For transformations, pick a Transformation, change Rotation angle about O or Dilation ratio k, and drag A, B, C or the tip of vector v while comparing the coordinates of each point and its image. For graphs, choose a Sample graph, count the odd-degree vertices, then click Find Eulerian path to see the edges coloured in order. For drawing, choose the Solid to project and drag sideways to rotate it beside its three views.

Parameters you can change

• Current topic 1. Transformations in the plane, 2. Introduction to graph theory, 3. Elements of technical drawing
• Transformation Translation by vector v, Rotation about center O, Dilation about center O, Reflection across the Ox axis
• Rotation angle about O -180–180 deg
• Dilation ratio k -2–3
• Sample graph Seven Bridges of Königsberg, Graph with an Eulerian path, Graph with an Eulerian circuit
• Solid to project L-shaped block, T-shaped block, Grooved block (U-shaped)

Questions to explore

1. Which of the four transformations keep the distance between any two points unchanged, and which do not?
2. Why is it impossible to cross all seven bridges of Königsberg exactly once each?
3. How does the side view of the T-shaped block differ from its front view?