Orthoptic (geometry)¶
The locus of points from which two tangents to a given curve meet at a right angle.
Core Idea¶
Only points admitting the required real tangents belong, singular and interior cases need conventions, the orthoptic is the ninety-degree member of the broader isoptic family and may be empty or disconnected. For a variable external point, tangent-contact equations determine two tangent directions; imposing zero dot product or a right-angle slope relation eliminates the contact parameters and yields the locus equation. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Orthoptic (geometry) belongs to plane geometry and is useful where the analyst can specify the typed plane geometry carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the source plane curve and regularity domain, observation point, two real tangent lines and their contact points, angle convention, orthogonality condition, elimination or envelope derivation, resulting locus and excluded singular or degenerate points, empty and complex cases and generalization to fixed-angle isoptics are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the source plane curve and regularity domain, observation point, two real tangent lines and their contact points, angle convention, orthogonality condition, elimination or envelope derivation, resulting locus and excluded singular or degenerate points, empty and complex cases and generalization to fixed-angle isoptics are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Orthoptic (geometry). Orthoptic (geometry) compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed plane geometry carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the source plane curve and regularity domain, observation point, two real tangent lines and their contact points, angle convention, orthogonality condition, elimination or envelope derivation, resulting locus and excluded singular or degenerate points, empty and complex cases and generalization to fixed-angle isoptics are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of plane geometry because they reuse the typed plane geometry carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, For a variable external point, tangent-contact equations determine two tangent directions; imposing zero dot product or a right-angle slope relation eliminates the contact parameters and yields the locus equation., and type the carrier, state every parameter and convention in the definition, test that the source plane curve and regularity domain, observation point, two real tangent lines and their contact points, angle convention, orthogonality condition, elimination or envelope derivation, resulting locus and excluded singular or degenerate points, empty and complex cases and generalization to fixed-angle isoptics are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Orthoptic (geometry) Domain-specific
Parents (1) — more general patterns this builds on
-
Orthoptic (geometry) is a kind of Relation Prime
The proposed strict upward parent is
prime:relation.
Hierarchy path (1) — routes to 1 parentless root
- Orthoptic (geometry) → Relation
Neighborhood in Abstraction Space¶
Orthoptic (geometry) sits in a crowded region of the domain-specific corpus (16th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Metric Geometry & Transformations (46 abstractions)
Nearest neighbors
- Cissoid — 0.92
- Curve — 0.92
- Tangential quadrilateral — 0.92
- Helix — 0.92
- Parabola — 0.91
Computed from structural-signature embeddings · 2026-09-08