Geometric Bites

Barycentric Coordinates Made Clear: From a UV Triangle to a 3D Triangle This post explains, in plain language, why a simple triangular region in a 2-dimensional parameter space can describe every point of a real triangle in 3-dimensional space. Through one clean affine formula, the 2D parameters (u, v) determine points on a 3D triangle. Once this connection is seen, concepts such as interpolation, texture mapping, geometric modelling, and FEM become much easier to understand. Two Worlds Connected by a Map 1) Parameter World (UV-space, 2D) Start in a flat coordinate plane labelled by two parameters, u and v . Consider the square defined by 0 ≤ u ≤ 1 0 ≤ v ≤ 1 If we cut this square along the diagonal line u + v = 1 (or equivalently v = −u + 1 ), we keep only the triangular region 0 ≤ u ≤ 1 0 ≤ v ≤ 1 u + v ≤ 1 This triangle has vertices (0,0) , (1,0) , (0,1) and is called the standard 2-simplex . Every point inside it is a convex mixture of its three corners. 2...

What Are Barycentric Coordinates? Barycentric coordinates provide a way to describe any point inside a triangle using the triangle’s own vertices as a reference. Instead of relying on the usual x–y axes, we express a point as a weighted combination of the three corners. The Core Idea Let a triangle have vertices A , B and C . Any point P inside (or on) the triangle can be written as P = αA + βB + γC The numbers α, β and γ are the barycentric coordinates of P. They indicate how strongly each vertex contributes to P. For this expression to make geometric sense, the three weights must satisfy α + β + γ = 1 This condition ensures that P behaves like a weighted average — a “blend” of A, B and C — rather than drifting away from the triangle. As long as all three values are non-negative ( α, β, γ ≥ 0 ), the point lies somewhere within the triangle. Examples α = 1 , β = γ = 0  →  P = A α = β = 0.5 , γ = 0  →  midpoint of AB α = ...

📐 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...

🧭 A Primer for Cross Product Calculations Right Hand Rule for Cross Product. Source: https://commons.wikimedia.org/wiki/File:Right-hand_rule_for_cross_product.png 1. Basis Vectors î = (1, 0, 0), ĵ = (0, 1, 0), k̂ = (0, 0, 1) These are unit and mutually perpendicular vectors. 2. Dot Product (for reference) 𝐀 · 𝐁 = |𝐀| |𝐁| cos θ î · ĵ = ĵ · k̂ = k̂ · î = 0 î² = ĵ² = k̂² = 1 3. Definition of the Cross Product 𝐀 × 𝐁 = |𝐀| |𝐁| sin θ n̂ θ is the angle from 𝐀 to 𝐁. n̂ is a unit vector perpendicular to both 𝐀 and 𝐁. The direction of n̂ follows the right-hand rule (anticlockwise = positive, clockwise = negative). 4. Fundamental Basis Cross Products î × ĵ = k̂ ĵ × k̂ = î k̂ × î = ĵ Reversing the order changes the sign: ĵ × î = −k̂ k̂ × ĵ = −î î × k̂ = −ĵ And any vector crossed with itself is zero: î × î = ĵ × ĵ = k̂ × k̂ = 0 5. Expansion in Component Form 𝐀 = a₁ î + a₂ ĵ + a₃ k̂ 𝐁 = b₁ î + ...