Coordinates after Descartes: geometry becomes algebra (3D)
About this app
A curve as a picture and as an equation at once: line, circle, parabola and ellipse on a 3D coordinate board, the equation as an animated transformation. Point test with P (on, inside, outside), intersection points with a line by substitution, discriminant and the quadratic formula, oblique axes as in Descartes (mixed term with cos ω) and the step into space: without z a cylinder and a plane, with z a sphere, ellipsoid and paraboloid.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Geometry becomes algebra
A coordinate system assigns a pair of numbers (x, y) to every point of the plane. A curve thereby becomes an equation: it consists exactly of the points whose coordinates satisfy the equation. By Pythagoras’ theorem, the circle around M(a, b) with radius r has the equation (x − a)² + (y − b)² = r². Geometric questions turn into calculations, for example finding intersection points: substitute the equation of the line into the equation of the curve and solve the resulting equation. René Descartes described this method in 1637 in “La Géométrie”; Pierre de Fermat found it independently at the same time.
What you see
A line, circle, parabola or ellipse lies on a 3D coordinate board, and next to it stands its equation as an animated transformation. You can place and drag a point P; the point test substitutes its coordinates and shows whether it lies on the curve, inside or outside. With a second line g the app computes the intersection points step by step: substitute, collect terms, discriminant, quadratic formula. The “Descartes” view slants the y axis, the “Space” view adds a third axis z.
Things to try
Put P on the circle around M(1, 1) with radius 2.5, for instance at (2.5, 3): both sides of the equation give 6.25. One step to the right and the left side grows; P lies outside. Switch on the line g: y = 0.5x + 1 and let the app compute the intersection points: S₁(3, 2.5) and S₂(−1.4, 0.3). Move g up until the discriminant turns negative. Then set the axis angle to 60° and watch the same curve get a longer equation, or switch to space: without z the circle becomes a cylinder, with z a sphere.
Frequently asked questions
- How do you check whether a point lies on a curve?
- Substitute the coordinates of the point for x and y in the equation of the curve. If both sides agree, the point lies on the curve. For a circle the inequality tells you more: if the left side is smaller than r², the point lies inside; if it is larger, outside.
- How do you compute the intersection points of a line and a circle?
- Substitute the equation of the line, y = mx + c, for y in the equation of the circle. This gives a quadratic equation in x. Its discriminant decides: if it is positive there are two intersection points, if it is zero the line touches the circle, if it is negative the line misses it. The equation of the line then gives the y values.
- Why are perpendicular coordinates called Cartesian?
- They are named after René Descartes, Latinised as Cartesius. In 1637, in “La Géométrie”, he showed how to solve geometric problems with equations. He did not yet draw a pair of axes with two scales; he measured lengths from a reference line at a fixed angle that did not have to be a right angle. The perpendicular pair of axes became established later.
- What does the app “Coordinates after Descartes” show?
- It shows a line, circle, parabola and ellipse as a picture and as an equation at the same time. You can place a point and test it, have the intersection points with a line computed, slant the axes and switch to space. An explain mode demonstrates the key steps.
Subject: Analytic geometry
More from Analytic geometry
- Point-to-line distance in 3D
- Distance between skew lines in 3D
- Planes in space
- Right-handed systems in 2D and 3D
- Shoelace method: the cross product
- Dot product: coordinate formula from the definition
- Cross and scalar triple products in 3D
- Angles between vectors in 3D
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