Geometric Bites

📐 What Is Graphjacking? Graphjacking is the creative act of using 2D graphing tools — such as Desmos or GeoGebra — to produce the illusion of 3D or higher-dimensional space. It turns a flat coordinate plane into a window for exploring depth, rotation, and perspective through pure mathematics. ⚙️ Definition Graphjacking is the process of taking a two-dimensional graphing system and manipulating equations to create 3D-like visualizations. It uses projection and trigonometric techniques to simulate a third dimension within the limits of a 2D plane. 🎨 Examples Drawing isometric cubes or dodecahedra on graph paper. Animating a rotating cube using trigonometric functions. Creating optical illusions such as the “Pringle surface.” 📚 Applications Education: Visualizing higher-dimensional concepts intuitively. Art: Designing 2D mathematical works that appear three-dimensional. Mathematics: Exploring projections, transformations, and geometry in creative wa...

Below are the workings you can use to find the intersection points of two quadratic equations . Part 1 Part 2   There is a free interactive graph related to these workings here: https://www.desmos.com/calculator/0twcp4hwx2

Below is an example of how to find the region where two circles on a 2 dimensional plane overlap . The workings are in high resolution... To demonstrate the workings are indeed correct, a free graph has been created on Desmos: https://www.desmos.com/calculator/0aonrev54r

In this post I demonstrate how to find out where a line and circle intersect on a 2 dimensional plane. The first step is to write out the equations of a line and a circle. We then make sure for both equations 'y' is isolated . Once we have the equations for a line and a circle whereby 'y' is isolated, we can then go about finding the values of 'x' for where the line intersects the circle. Like so... After we have found the values of 'x' using the quadratic formula , we then plug them back in to y=mx+c to get the values of 'y' for which the line intersects the circle. And that's it basically. Below is a free interactive Desmos graph related to this work: https://www.desmos.com/calculator/w3tsnjajtx

Below are the workings required to build an interactive graph that finds the points where a line and quadratic equation intersect . Here 'a' is not equal to 0 as it becomes part of the denominator a fraction . Also, if 'a' were equal to 0, we'd have a line intersecting a line, not a parabola (for instance). When we have the formula for 'x', we just plug it back into the linear equation to get the outcomes for 'y'. And there we have it, the points where a line and quadratic equation intersect. To see the interactive graph I was talking about, visit the Desmos link below: https://www.desmos.com/calculator/sqzjajbhea