Sloped Frustum of a Pyramid or Cone
Projection of Base to Sloped Plane
Hip Run (Cone Radius) =
Hip (Cone) Slope Angle =
Cutting Plane Angle =
Section Plane Rotation Angle =

Nodes: Pyramid Base Circumcircle

x =
± y =
Radius – y =

Origin at Conic Section
intersection with Circle Center

Type of Conic Section

Section Slope Angle =
Frustum Surface Angle =
Frustum Radius Vector =
on frustum surface ... ± Ö(y ² + z ²) =
in plan view ... ± Ö(x ² + y ²) =
on line of intersection ... x =
plan view ... y =
height or rise ... z =
Pythagorean ... Arc Length =
Simpson ... Arc Length =
Sheathing Angle =
Tangent to Conic Section =
Normal to Conic Section =

Origin at Conic Section Centroid

Major Axis =
Minor Axis =
Focus =
Conic Section Centroid – Circle Center =
x =
± y =
Minimum Bounding Radius =
Maximum Bounding Radius =

Spline ... the Conic Section in Plan View
Equation of the Spline Radius Vector
r = κε ÷ (1 + ε sin θ)

Eccentricity ... F/b = ε =
Focus to Directrix distance ... κ =
Spline Radius Vector ... r =
Spline Major Axis =
Spline Minor Axis =
Spline Focus =
Pythagorean ... Spline Arc Length =
Simpson ... Spline Arc Length =
Plan Angle =
Tangent to Spline =
Normal to Spline =

Links to related material
Sloped Frustum of a Pyramid ... Supplementary Diagrams
Sloped Frustum of a Pyramid ... Sleeper Blade Bevel Calculation
General Case ... Sleeper Blade Bevel Formulas
General Case ... Layover Rafter intersects slope of Main Roof

Tetrahedron equivalent to model of Hip Roof extracted from Frustum of Cone
Formulas for Hip Roof extracted from Frustum of Cone
Orthographic Projection of Sloped Frustum Geometry
Conic Section ... Base of Cone projected to the Cutting Plane
Section through Cone ... positive offset Cone and Cutting Plane Intersection
Section through Cone ... positive offset Cone and Cutting Plane Intersection
Section through Cone and Spline in Plan View ... positive offset Intersection
Plan View ... positive offset Cone and Cutting Plane Intersection
Section through Cone ... negative offset Cone and Cutting Plane Intersection
Section through Cone ... negative offset Cone and Cutting Plane Intersection
Section through Cone and Spline in Plan View ... negative offset Intersection
Development of Valley Ellipse
Development of Valley Ellipse
Focus-Directrix Equation of the Spline ... the Spline is an Ellipse
Polar Equations of the Spline
Model of Spline Ellipse and Valley Ellipse

Front View ... Developments of Spline Ellipse and Valley Ellipse

Development of Spline and Valley Elliptical Arc Lengths

Back View ... Developments of Spline Ellipse and Valley Ellipse
Main Side Jack Purlins intersect Curved Valley Beam
Adjoining Side Jack Rafters intersect Curved Valley Beam
Joe Bartok