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# The Problem of Regiomontanus

[GitHub](https://github.com/HectorTablero/QED-Scripts/tree/main/Magazine%202/The%20Problem%20of%20Regiomontanus) (Open Source, MIT License) | [Live Demo](https://qed.mat.uam.es/revista/articulo/regiomontano)

**Date:** July 2026
**Technologies:** JavaScript, Tailwind CSS

---

## Overview

The article reconstructs how Regiomontanus could have solved, in the 15th century and without calculus, the classic problem of the optimal viewing distance from a painting hung high on a wall: at what distance $x$ is the angle subtended by the painting largest? The solution route runs through two synthetic-geometry lemmas — a diagram proof of the sine addition formula, and a compass-and-straightedge proof of the chain of means $HM \le GM \le AM \le RM$ — before arriving at the answer $x = \sqrt{ab}$. Three elements let the reader manipulate each stage directly: the viewing problem itself, the sine-sum construction, and the means-inequality construction.

## RegiomontanusAngle

### Description

A draggable gallery scene where the reader positions a viewer at distance $x$ from the wall and watches the angle $\alpha - \beta$ subtended by the painting update live, alongside a graph of that angle as a function of $x$ tracking the current position.

### Implementation

```javascript
// Example element configuration
::: js RegiomontanusAngle
3, 1.2, 0.8
regiomontano1
autoPlay, allowFullscreen, openControls, h=225
:::
```

Parameters are `[a, b, x]`: the heights of the painting's top and bottom edges above eye level, and the initial viewing distance. Dragging the viewer along the floor (`pointerdown`/`pointermove` on the lower portion of the scene) updates $x$ and both panels simultaneously.

### The Optimum and the Degenerate Case

A "show the optimum" toggle overlays $x = \sqrt{ab}$ as a vertical line in both panels, plus the classical circle through the painting's two corners that is tangent to the floor at exactly that point — the synthetic-geometry object the article's solution is built around. When `b <= 0` (the painting's bottom edge is below eye level), the element switches to reporting the article's own noted degenerate case: the optimal angle keeps increasing all the way to $x=0$, i.e. "press your nose to the picture," which the readout states explicitly rather than leaving the reader to discover a graph that never turns over.

### Mathematical Background

The viewer at distance $x$ from the wall sees the top edge at elevation angle $\arctan(a/x)$ and the bottom edge at $\arctan(b/x)$, so the subtended angle computed by `view(x)` is exactly

$$\theta(x) = \arctan\!\left(\frac{a}{x}\right) - \arctan\!\left(\frac{b}{x}\right).$$

A calculus derivation (the one the pre-calculus article deliberately avoids) maximizes this by differentiating: $\theta'(x) = \dfrac{-a}{x^2+a^2} - \dfrac{-b}{x^2+b^2} = 0$ gives $b(x^2+a^2) = a(x^2+b^2)$, i.e. $(a-b)x^2 = ab(a-b)$, so for $a \ne b$ the critical point is $x=\sqrt{ab}$ — the geometric mean of the two heights, matching the overlay. Equivalently, since $\theta$ is the difference of two arctangents, $\tan\theta = \dfrac{a/x - b/x}{1+ab/x^2} = \dfrac{(a-b)x}{x^2+ab}$, and this ratio is itself maximized (by AM-GM on $x$ and $ab/x$) exactly when $x = ab/x$, i.e. $x=\sqrt{ab}$, without any derivative at all — the synthetic route the tangent circle encodes: the circle through the two corners tangent to the eye-line touches it precisely where the viewing angle peaks, a classical fact about the angle subtended by a fixed chord being maximal at the tangency point of the inscribed circle.

## SineSumDiagram

### Description

Renders the specific compass diagram the article uses to prove $\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ synthetically, with $\alpha$ and $\beta$ on independent sliders and the two component lengths $q = \cos\alpha\sin\beta$, $r = \sin\alpha\cos\beta$ labeled as measured segments so that $\text{AE} = r + q$ is watched holding rather than asserted.

### Implementation

```javascript
// Example element configuration
::: js SineSumDiagram
35, 28
senosuma1
autoPlay, allowFullscreen, openControls, h=195
:::
```

Parameters `[alphaDeg, betaDeg]` set the initial angles in degrees. Both sliders are mutually capped so that $\alpha + \beta < 90°$ (`cap = 87` in the source), since outside the first quadrant the diagram's right triangles and semicircle construction stop corresponding to the identity being illustrated.

### Construction

The element draws all six labeled points ($O, A, B, D, E, F$) from the article's text exactly as described — three rays from $O$, a semicircle on diameter $OA$, and orthogonal projections onto the base ray — and colors the three segments $q$, $r$, and $\text{AE}=\sin(\alpha+\beta)$ distinctly (red, blue, green) so their sum relationship is legible directly from the figure, with a live numeric readout confirming the identity to four decimal places for the current angle values.

### Mathematical Background

$O$ is the origin, $A=(\cos(\alpha+\beta),\sin(\alpha+\beta))$ sits on the unit circle at angle $\alpha+\beta$, and $B=(\cos\beta\cos\alpha,\ \cos\beta\sin\alpha)$ is the point at distance $\cos\beta$ along the ray at angle $\alpha$ (i.e. $OB=\cos\beta$, since $B$ is the foot of the perpendicular from $A$ onto that ray, using the Thales/semicircle property that $\angle OBA = 90°$). Dropping $D$, $E$, $F$ as the orthogonal projections shown in the code, the segment $\text{AE}$ — the height of $A$ above the base ray — decomposes as $\text{AE} = \text{AF} + \text{FE} = \text{BD} + \text{BF}$, and since triangle $OBD$ gives $\text{BD} = OB\sin\alpha = \cos\beta\sin\alpha$ while the right triangle at $B$ gives $\text{BF} = AB\cos\alpha = \sin\beta\cos\alpha$ (as $AB=\sin\beta$, the leg opposite $\beta$ in the small right triangle on hypotenuse $OA=1$), the sum is precisely the sine addition formula:

$$\sin(\alpha+\beta) = \underbrace{\sin\alpha\cos\beta}_{r,\ \text{blue}} + \underbrace{\cos\alpha\sin\beta}_{q,\ \text{red}},$$

with $\text{AE}$ (green) drawn as their sum $r+q$ rather than computed independently, which is exactly why the readout "confirms" rather than assumes the identity.

## MeansInequalityConstruction

### Description

Reconstructs the compass diagram proving $HM \le GM \le AM \le RM$ (harmonic, geometric, arithmetic, and quadratic means) for two values $a \ge b$, pairing the geometric figure with a bar chart of the four resulting lengths.

### Implementation

```javascript
// Example element configuration
::: js MeansInequalityConstruction
4, 1
medias1
autoPlay, allowFullscreen, openControls, h=195
:::
```

Sliders for `a` and `b` are mutually constrained (`a = max(v, b)`, `b = min(v, a)`) so the diagram never has to handle $a < b$, and an "make a equal b" button drives the construction to its degenerate equality case directly.

### Construction and Bars

Following the article's own labeling, the element places $M$ at the origin and constructs the circle on diameter $PQ$, the tangent from $M$ meeting it at $G$, the perpendicular through the center $A$ meeting it at $R$, and the altitude foot $H$ — then draws the four segments $HM$, $GM$, $AM$, $RM$ in matching colors both in the geometric figure and as a synchronized bar chart, so the ordering claimed by the inequality is visible as both a nested-segment picture and a sorted bar height. As $b \to a$, the circle's radius shrinks to zero and all four bars converge, mirroring the article's own equality-case argument (a degenerate triangle forces $a=b$).

### Technical Considerations

All three elements are pure `Plot2D`-based canvas drawings with no external geometry library: circle tangency points, projections, and the optimal-viewing circle are all computed directly from closed-form trigonometric or Pythagorean expressions matching the article's synthetic proofs step for step, so the figures are derivations made visible rather than illustrative approximations.