Design and Communication Graphics · 6th Year · Applied Graphics: Structural Forms · 3.º Período

Hyperbolic Paraboloids

Students investigate the geometry of doubly ruled surfaces, specifically hyperbolic paraboloids. They draw these structures in plan, elevation, and end view.

TL;DR:Hyperbolic Paraboloids (Hypars) are among the most striking structural forms in modern architecture, famous for their 'saddle' shape. In the DCG syllabus, they are studied as 'doubly ruled surfaces,' meaning that even though the surface is curved, it is composed entirely of straight lines (generators). This unique geometric property makes them both beautiful and incredibly strong, as seen in the roof of the Scotiabank Saddledome or various modern Irish church designs.

• NCCA Curriculum Specifications
• NCCA DCG Syllabus Applied 2.1: Structural Forms
• NCCA DCG Syllabus Core 1.1: Orthographic Projection

About This Topic

Hyperbolic Paraboloids (Hypars) are among the most striking structural forms in modern architecture, famous for their 'saddle' shape. In the DCG syllabus, they are studied as 'doubly ruled surfaces,' meaning that even though the surface is curved, it is composed entirely of straight lines (generators). This unique geometric property makes them both beautiful and incredibly strong, as seen in the roof of the Scotiabank Saddledome or various modern Irish church designs.

Students must learn to project these forms in plan and elevation, and more importantly, to find the true shape of sections cut through them. This topic challenges a student's ability to manage a large number of points and lines without losing track of the overall form. It is a masterclass in the power of 'ruled geometry.'

This topic comes alive when students can physically model the patterns using thread and a frame, seeing how straight lines magically create a curved surface.

Key Questions

1. What defines a doubly ruled surface?
2. How are hyperbolic paraboloids used in modern architecture?
3. How do we determine the true shape of a section through a hypar?

Watch Out for These Misconceptions

Common Misconception

Students often think that because the surface is curved, the lines (generators) must also be curved.

What to Teach Instead

Use the string model activity. When students pull a string tight between two points, they see it is perfectly straight, yet the overall shape is curved. This 'aha!' moment is crucial for their understanding of ruled surfaces.

Common Misconception

Confusion between the two sets of generators.

What to Teach Instead

Use two different colors of thread in the modeling activity, one for the 'first generation' and one for the 'second generation.' This makes it clear that every point on the surface lies on two different straight lines.

Active Learning Ideas

Inquiry Circle

The String Model

In small groups, students build a simple wooden or card frame. They then use colorful thread to create a hyperbolic paraboloid by connecting points at equal intervals along the frame. This physical model serves as a reference for their orthographic drawings.

50 min·Small Groups

Think-Pair-Share

The Plane Section Challenge

Show a Hypar being cut by a horizontal plane. Students individually predict the shape of the resulting section (is it a parabola or a hyperbola?). They then pair up to construct the section on paper and verify their prediction.

30 min·Pairs

Gallery Walk

Hypars in Architecture

Display photos of famous Hypar buildings around the world. Students move in pairs to identify the 'directrices' and 'generators' in each photo, marking them on transparent overlays to see how the geometry translates to real steel and concrete.

25 min·Pairs

Frequently Asked Questions

What is a 'doubly ruled surface'?

A doubly ruled surface is a surface where through every point, there are two distinct straight lines that lie entirely on the surface. The hyperbolic paraboloid and the hyperboloid of revolution are the two main examples studied in DCG.

Why are Hypars used in roof design?

They are incredibly efficient. Because they are made of straight lines, they can be built using straight timber or steel beams, yet they provide the structural strength of a curve. They also shed water very effectively due to their 'saddle' shape.

How can active learning help students understand Hypars?

Active learning, specifically physical string modeling, is the most effective way to demystify Hypars. It transforms an abstract, complex drawing into a tangible object. Once a student has built a Hypar, they no longer struggle to understand why we use straight lines in the plan and elevation to represent a curved roof.

How do you find the true shape of a section through a Hypar?

You use the same principles as any other solid: project the points where the cutting plane hits the generators into an auxiliary view. The key is staying organized and labeling your generators (A, B, C...) clearly in both views.