F.54 HP-35s Triangle Program and Coordinate Scripts

BARY 2 CAR and CAR 2 BARY Matrix Routines, Register Map A–Z, and 5-6-7 Verification

In the canonical frame A = (0, 0), B = (M, 0), C = (I, J), the forward and inverse barycentric-Cartesian transformations BARY 2 CAR and CAR 2 BARY are given by P = M U + I V, Q = J V and V = Q / J, U = (P - I V) / M, T = 1 - (U + V).

Interactive GEOMETOR construction for F.54 (HP-35s Triangle Program and Coordinate Scripts). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point P : Point ===== s_ab = segment(A, B) H_C = foot(C, line(A, B)) H_P = foot(P, line(A, B))

Mathematical Synthesis

Dated September 10, 2014 with LAST EDIT May 7, 2015 (SeeligNotebooks), these two folios document Seelig’s HP-35s [T] [Triangle] calculator program specification. Placing A at \((0,0)\), B at \((M,0)\), and C at \((I,J)\), the program maps normalized homogeneous barycentric coordinates \((T, U, V) = (\lambda_A, \lambda_B, \lambda_C)\) to Cartesian coordinates \((P, Q) = (x, y)\) via the upper-triangular routine BARY 2 CAR: \(P = MU + IV, Q = JV\).

Conversely, the inverse routine CAR 2 BARY recovers the barycentric weights in three register operations: \(V = Q/J\), \(U = (P - IV)/M\), and \(T = 1 - (U + V)\). Folio 2 records the complete A–Z memory register map of the HP-35s alongside a worked validation on the canonical \((5,6,7)\) triangle.

Key Algebraic Identities

\[\begin{split}\text{BARY 2 CAR:}\quad \begin{bmatrix} P \\ Q \end{bmatrix} = \begin{bmatrix} M & I \\ 0 & J \end{bmatrix} \begin{bmatrix} U \\ V \end{bmatrix}, \qquad T + U + V = 1\end{split}\]

\[\text{CAR 2 BARY:}\quad V = \frac{Q}{J}, \qquad U = \frac{P - I V}{M}, \qquad T = 1 - (U + V)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.54 HP-35s Triangle Program and Coordinate Scripts — Folios (Folios 0347–0348, 2 scanned manuscript pages in resources/seelig/54_scripts)
Euclid’s Elements	VI.2 (EuclidElements)
GEOMETOR Elements (G-Index)	G.5 surface
Encyclopedia of Triangle Centers	X(1) — Incenter, X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter (Kimberling1998)
Seelig Spreadsheets	Barycentric, 5-6-7 A,B,C (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.47 The UVW Coordinate Triangle, F.52 Homogeneous Barycentric Coordinates