Research

Aesthetic Curves ⊂ Geometric Modelling

I have been working on geometric modelling since my undergraduate days, with a major in applied mathematics and minor in fine arts from USM Penang. Been always interested with shapes, curious about what kind of shapes excites emotion. One key feature of shapes is the rate of bending / twisting in space which can be represented with a curvature and torsion function, respectively. Hence, i have spent ample of time working on especially, curvature functions. The implication of curvatures is exceptionally vast, you need to inspect the curvature of the shape that you're designing to avoid unnecessary dents. It is also essential for robotic path design and highway design.

A direct experience of curvature is when you drive a car on a highway. The way you turn your car steering says it all; if you keep turning left, then the curvature is positive, whereas if you turn to the right, then it is negative. If you are in a straight road, then you do not turn of course! hence the curvature is zero. An inflection point is when the curvature changes sign, e.g. turning left, then to right. When you drive on a highway, you would experience a gradual 'turning' experience. You are in fact turning a linear form; simply means the curvature is a linear function and the curve which naturally has linear curvature is called clothoid or Cornu spiral. This spiral has been used as a template for highway design for a long time now.

Curvature is the second derivative of the curve itself. Thus, when you integrate curvature, you get the turning angle, means the tangent of your traverse. You can proceed to integrate again to get the curve. The process of deriving curve from curvature is known as curve synthesis; as the wording goes, you 'synthesise' a new curve by defining a curvature function. In the realm of geometric modelling and shape design, researchers are interested in finding new curves with 'good' properties to facilitate design process and at the same time produce visually pleasing shapes. In this context, we know that a monotonic curvature function will produce an aesthetic shape. So one way to search for a good candidate is to search for monotonic functions, however this can be an exhausting process.

Image you collect all the curves from natural products and aesthetic man-made products and figure out its curvature representation. This is how we landed ourselves working with Log-Aesthetic curves (LAC) for some time now. The beauty of this curve is that it has a self-affine characteristic; you can choose a part of the curve and transform it to represent the original curve. Thus fitting with G1 Hermite data (end positions and its tangent directions) can be done in few steps. However, fitting G2 data (G1 data plus its curvatures) can be tricky due to scaling process which changes the curvature values. One way to deal with G2 data is by segmenting the curve to increase degree of freedom. It is unknown how self-affine characteristic is helpful for design or manufacturing process, but what we know is that it can be related to the beauty of self similar objects. Self similarity is a special case of self affinity and well known for fractal studies.

Figure in the middle shows a logo designed with four-point Generalized Log-Aesthetic curves (GLAC) with tangent continuity. Figure on the left shows obstacle avoiding curvature continuous bi-LAC minimises given metrics. Clothoid (brown curve) fails to avoid given obstacle and G1 data, whereas with LAC (red) you have an extra shape variable which can be utilised to meet the design constraints, e.g. minimise arc length, bending energy or curvature variation energy at the same time avoid the obstacles.

With the collaboration of six Japanese universities, LAC is now extensively explored for various kind of applications; details can be found at CREST-ED3GE.

What you don't see, doesn't mean you can't comprehend!

We are now overwhelmed with enormous data, trying to filter whats is essential by asking various questions. What form does it take? Linear or qubic? Number of connected components, number of loops, voids it has, or number of clusters... In short, what we are seeking is the shape of this high dimensional dataset. It is unfortunate that we cannot visualise the shape in n-dimension since we humans are 3D centric. A workaround would be to reduce their dimensionality and figure out its charateristics, e.g., using principal component analysis (PCA) or multidimensional scaling (MDS). However, we may loose some information when we reduce the dimension dataset.

Crunching n-dimensional dataset is doable with Topological Data Analysis (TDA); it is coordinate free, invariance under deformation and sensitive to large & small scale patterns. The underlying principle of TDA is from set theory, topology, combinatorics, modern and linear algebra. Oh yes, it has some hardcore mathematics which might scare newbies away. However, it was pretty natural extending geometric modelling work from 2-dimension to n-dimension in order to comprehend its shape.

"Data has shapes" is a selling point of TDA for data science community. It is vastly used to explore various types of dataset before concentrating on a particular method to deduce, thus TDA can be classified as exploratory data analysis tool. It sure intrigued me back in 2017 while browsing some articles. In summer 2018, I attended a seminar at The Hamilton Mathematics Institute (HMI) at Trinity College Dublin, presenting a work on LACs. I was fortunate to attend four hour lecture by John Harer, the author of "Computational Topology: An Introduction". I was amazed with the details and pretty excited what it has to offer. Back to office, i kept gathering information TDA whenever i had free time. Back in Malaysia, Salmi Noorani (UKM) and his students has already embarked in TDA using R Package. In Sept 2018, he gave a talk in USM during a workshop and was happy to discuss with his students' TDA journey.