5.2 Arcs, Circles, Ellipses & Splines

5.2 Arcs, Circles, Ellipses & Splines

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1. ARC Construction Methods & Mathematical Rules

AutoCAD provides 11 distinct methods for constructing circular arcs via the ARC (or A) command. Understanding how AutoCAD interprets start points, centers, angles, directions, and radii is critical for rapid geometry creation.

AutoCAD Arc Direction Convention:

        CCW Direction (Default)                   CW Direction (Hold Ctrl Key)
             ◄────────┐                                 ┌────────►
                      │                                 │
           Start ─────┴─────► End            Start ─────┴─────► End

Arc Method Breakdown

Mathematical Limitations of Chord Length (S, C, L)

In geometry, a chord is the straight line segment joining two points on a curve. The longest possible chord in any circle is its diameter ($2 \times R$). If a drafter enters a chord length value that exceeds the diameter ($L > 2R$), AutoCAD rejects the input because no such arc exists.

• Specifying a positive chord length draws the minor arc (included angle $\le 180^\circ$).
• Specifying a negative chord length draws the major arc (included angle $> 180^\circ$).

Minor vs. Major Arc Resolution in Start-End-Radius (S, E, R)

When connecting two points with a fixed radius, two possible circular arcs can bridge the gap: a minor arc (spanning $\le 180^\circ$) and a major arc (spanning $> 180^\circ$):

• Minor Arc: Enter the radius as a standard positive number (e.g., 50).
• Major Arc: Enter the radius as a negative number (e.g., -50).

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2. CIRCLE and ELLIPSE Construction Paradigms

The CIRCLE Command

The CIRCLE (or C) command offers six construction options:

1. Center, Radius: Click a center coordinate, then enter the radius value or pick a distance.
2. Center, Diameter (D): Prompts for total diameter rather than radius.
3. 2-Point (2P): Prompts for two points that define the exact diametrical ends of the circle.
4. 3-Point (3P): Constructs a circle passing through three arbitrary non-collinear boundary points.
5. Tan, Tan, Radius (TTR): Requires selecting two tangent references (lines, arcs, or circles) followed by a radius value. If the radius is mathematically too small to reach both objects or too large to fit between them, AutoCAD displays "Circle does not exist".
6. Tan, Tan, Tan (TTT): Accessible via the Ribbon (Home > Draw > Circle > Tan, Tan, Tan), this executes the 3-Point circle command with automated Deferred Tangent (_tan) OSNAP overrides applied to each prompt.

The ELLIPSE Command and PELLIPSE System Variable

An ellipse is defined by a major axis (longest diameter) and a minor axis (shortest diameter). The ELLIPSE (or EL) command provides three creation modes:

• Axis, End: Defines the length and orientation angle of the first axis by its two endpoints. The second axis is then defined by specifying the perpendicular half-length distance from the axis midpoint to the ellipse boundary (or typing R to specify a 3D projection Rotation angle).
• Center (C): Establishes the ellipse center first, followed by the endpoint of the first axis, and then the distance to the second axis.
• Elliptical Arc (A): Constructs a partial ellipse. Prompts for axis parameters, followed by a Start angle and an End angle (or Included angle). Angles are calculated from the major axis in a CCW direction.

ELLIPSE Database Architecture:

PELLIPSE = 0 (Initial value):      ──► Creates a true ELLIPSE object
                                       Keeps the exact center, axes, and eccentricity

PELLIPSE = 1 (Legacy Polyline):    ──► Creates closed LWPOLYLINE
                                       Approximates the ellipse with arc segments

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3. Non-Uniform Rational B-Splines: The SPLINE Command

A SPLINE is a smooth curve generated using mathematical Non-Uniform Rational B-Splines (NURBS). Splines can represent complex ergonomic profiles, aerodynamic contours, boat hulls, and civil contour grading that cannot be drawn using standard circular arcs.

AutoCAD creates splines using two fundamentally different methods toggled via the Method (M) option:

SPLINE Creation Methods:

Fit Points Method:             [P1] ●───────● [P2]───────● [P3]───────● [P4]
                               (Curve passes directly THROUGH user-clicked points)

Control Vertices (CV) Method:        [CV2] □───────────────□ [CV3]
                                    ╱                       ╲
                                [CV1] ●                     ● [CV4]
                               (Curve is attracted to CV hull; does NOT pass through CV2/CV3)

Spline Degree

The Degree option defines the polynomial degree ($D$) of the NURBS curve equation:

• Valid values range from 1 to 10 (Default is 3, representing a standard cubic B-spline).
• Degree 1: Yields straight linear segments between vertices (behaves like a polyline).
• Degree 2: Quadratic spline; highly stable with smooth slope continuity ($C^1$).
• Degree 3 (Cubic): Standard production default; provides continuous slope and curvature continuity ($C^2$).
• Degrees 4–10: Higher-order curves providing higher mathematical smoothness ($C^3$ to $C^9$) for specialized automotive and aerospace surface styling.

Knot Parameterization

When creating splines using Fit Points, AutoCAD must assign mathematical parameters (knots) to each segment between data points. The Knot (K) option offers three mathematical parameterization schemes:

1. Chord: The knot spacing is proportional to the straight-line Euclidean distance between consecutive fit points. This is the industry-standard general-purpose setting and handles non-uniformly spaced data points effectively.
2. Square Root (Centripetal): The knot spacing is proportional to the square root of the distance between consecutive points ($\sqrt{\text{distance}}$). This method prevents overshoot, cusps, and self-intersecting loops when fit points make sharp, abrupt directional changes.
3. Uniform: Knots are spaced at equal integer intervals ($0, 1, 2, 3\dots$) regardless of physical point spacing. Uniform parameterization is only appropriate when fit points are already spaced at perfectly equal intervals; if data points vary in distance, Uniform produces extreme, unwanted undulations and wiggles.

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4. Advanced Spline Modification: SPLINEDIT

The SPLINEDIT (or SPE) command provides comprehensive tools for interrogating and refining NURBS geometry.

Core SPLINEDIT Options

• Close / Open: Seamlessly joins the start and end points of an open spline, enforcing continuous curvature ($C^2$) across the closure seam.
• Fit Data (F): Available only on splines that retain fit point history. Allows drafters to:
  • Add: Insert additional fit points.
  • Delete: Removes selected fit points (the curve recalculates through the remaining points).
  • Kink: Adds a knot and fit point that does not keep tangent or curvature continuity at that point, creating a sharp corner.
  • Move: Reposition fit points to new coordinates.
  • Purge: Permanently strips fit point data from the drawing database, converting the spline into a pure Control Vertices (CV) representation to save memory.
  • Tangents: Redefines start and end tangent vectors.
  • Tolerance: Adjusts the fit tolerance smoothing band.

The Edit Vertex Options

Selecting Edit vertex exposes the control-frame editing options:

• Add: Adds a control vertex at a point you specify between two existing control vertices, giving more local control.
• Delete: Removes a selected control vertex.
• Elevate Order: Increases the polynomial order (degree plus one) of the spline, which adds control vertices across the whole curve. The maximum order is 26.
• Move: Relocates a selected control vertex.
• Weight: Changes the weight of a control vertex. A larger value pulls the spline closer to that vertex; a smaller value lets the curve fall away from it.

Convert to Polyline and PLINECONVERTMODE

Splines can be converted into 2D polylines using the Convert to Polyline (C) option within SPLINEDIT:

Command: SPLINEDIT
Select spline: [Select Spline]
Enter an option: Convert to Polyline
Specify a precision <10>:

AutoCAD prompts for a Precision value between 0 and 99. The higher the precision value, the closer the polyline matches the spline, at the cost of generating more vertices and slower performance. The DELOBJ system variable decides whether the original spline is kept.