Personally, I think the best way to illustrate curvature is to start from the simplest case. This is how Hilbert illustrated it in his book, Geometry and the Imagination.

1. The simplest curve is the the straight line and its curvature obviously should be zero.

2. The next simplest curve is the circle. Since the only invariant of a circle is the radius, the curvature must involve this. The larger the circle, the less curved it is. The simplest way to describe this is to say that curvature is the reciprocal of the radius.

3. For any other curve we measure its curvature by fitting a circle to it and calling its radius, the radius of curvature.

Of course this leaves unexplained how to actually find this fitting circle. Both Newton and Liebniz provided different but equivalent method: Basically, take two points on either side of the chosen point and draw the tangent there. Then draw the normals. Where they intersect is the approximate centre of this circle. Then by taking the limits of both points approaching the chosen point we find the true centre and hence radius of curvature.

I like this because it directly connects our intuitive notion of curvature to its formalisation. Hilbert doesn't describe how to generalise this to higher dimensions, but presumably it's a question of fitting ellipsoids...

For example, on a surface, we would get a fitting ellipse whose major and semi-major axes will be the principal axes of curvature. Gauss then showed that the product of these the two radii, the Gaussian curvature, was independent of the embedding of the surface in Euclidean space - his theorem egregium, his 'remarkable theorem'. This inaugurated the era of intrinsic geometry as opposed to extrinsic geometry where we consider a geometrical object embedded in another.

Of course it is a mistake to think intrinsic geometry is the final word on geometry. After all, the theory of bundles (and of cobundles), as used in gauge theory and homology/cohomology shows that it is equally important to keep the extrinsic view in mind.

More practically, we can see this with the Möbius strip. Intrinsically, it can only have two twists; the trivial one and the half-twist; extrinsically, there is a natural numbers worth of full twists and of half twists. This is important in the theory of spin in physics.

Personally, I think the best way to illustrate curvature is to start from the simplest case. This is how Hilbert illustrated it in his book, Geometry and the Imagination.

1. The simplest curve is the the straight line and its curvature obviously should be zero.

2. The next simplest curve is the circle. Since the only invariant of a circle is the radius, the curvature must involve this. The larger the circle, the less curved it is. The simplest way to describe this is to say that curvature is the reciprocal of the radius.

3. For any other curve we measure its curvature by fitting a circle to it and calling its radius, the radius of curvature.

Of course this leaves unexplained how to actually find this fitting circle. Both Newton and Liebniz provided different but equivalent method: Basically, take two points on either side of the chosen point and draw the tangent there. Then draw the normals. Where they intersect is the approximate centre of this circle. Then by taking the limits of both points approaching the chosen point we find the true centre and hence radius of curvature.

I like this because it directly connects our intuitive notion of curvature to its formalisation. Hilbert doesn't describe how to generalise this to higher dimensions, but presumably it's a question of fitting ellipsoids...

For example, on a surface, we would get a fitting ellipse whose major and semi-major axes will be the principal axes of curvature. Gauss then showed that the product of these the two radii, the Gaussian curvature, was independent of the embedding of the surface in Euclidean space - his theorem egregium, his 'remarkable theorem'. This inaugurated the era of intrinsic geometry as opposed to extrinsic geometry where we consider a geometrical object embedded in another.

It's probably worthwhile to add that this product is - upto a constant of proportionality - the area of the bounding rectangle. And this hints at the usefulness of tensors in dealing with curvature as tensors geometrically represent volumes (that is, upto rotations and shears).

Of course it is a mistake to think intrinsic geometry is the final word on geometry. After all, the theory of bundles (and of cobundles), as used in gauge theory and homology/cohomology shows that it is equally important to keep the extrinsic view in mind.

More practically, we can see this with the Möbius strip. Intrinsically, it can only have two twists; the trivial one and the half-twist; extrinsically, there is a natural numbers worth of full twists and of half twists. This is important in the theory of spin in physics.

Personally, I think the best way to illustrate curvature is to start from the simplest case. This is how Hilbert illustrated it in his book, Geometry and the Imagination.

1. The simplest curve is the the straight line and its curvature obviously should be zero.

2. The next simplest curve is the circle. Since the only invariant of a circle is the radius, the curvature must involve this. The larger the circle, the less curved it is. The simplest way to describe this is to say that curvature is the reciprocal of the radius.

3. For any other curve we measure its curvature by fitting a circle to it and calling its radius, the radius of curvature.

I like this because it directly connects our intuitive notion of curvature to its formalisation. Hilbert doesn't describe how to generalise this to higher dimensions, but presumably it's a question of fitting ellipsoids...

Personally, I think the best way to illustrate curvature is to start from the simplest case. This is how Hilbert illustrated it in his book, Geometry and the Imagination.

1. The simplest curve is the the straight line and its curvature obviously should be zero.

2. The next simplest curve is the circle. Since the only invariant of a circle is the radius, the curvature must involve this. The larger the circle, the less curved it is. The simplest way to describe this is to say that curvature is the reciprocal of the radius.

3. For any other curve we measure its curvature by fitting a circle to it and calling its radius, the radius of curvature.

Of course this leaves unexplained how to actually find this fitting circle. Both Newton and Liebniz provided different but equivalent method: Basically, take two points on either side of the chosen point and draw the tangent there. Then draw the normals. Where they intersect is the approximate centre of this circle. Then by taking the limits of both points approaching the chosen point we find the true centre and hence radius of curvature.

I like this because it directly connects our intuitive notion of curvature to its formalisation. Hilbert doesn't describe how to generalise this to higher dimensions, but presumably it's a question of fitting ellipsoids...

For example, on a surface, we would get a fitting ellipse whose major and semi-major axes will be the principal axes of curvature. Gauss then showed that the product of these the two radii, the Gaussian curvature, was independent of the embedding of the surface in Euclidean space - his theorem egregium, his 'remarkable theorem'. This inaugurated the era of intrinsic geometry as opposed to extrinsic geometry where we consider a geometrical object embedded in another.

Of course it is a mistake to think intrinsic geometry is the final word on geometry. After all, the theory of bundles (and of cobundles), as used in gauge theory and homology/cohomology shows that it is equally important to keep the extrinsic view in mind.

More practically, we can see this with the Möbius strip. Intrinsically, it can only have two twists; the trivial one and the half-twist; extrinsically, there is a natural numbers worth of full twists and of half twists. This is important in the theory of spin in physics.