# What is a Gaussian Lobe concerning BRDF and NDF?

I was reading a paper about voxel cone tracing and came across this term. So, what is a Gaussian Lobe or a Lobe in general?

I am asking in the context of Computer Graphics, BRDF and NDF.

In a BRDF or NDF this most likely refers to a spherical Gaussian distribution, also known as a von Mises–Fisher distribution.

It's similar to the usual Gaussian, but using the dot product between two unit vectors as a measure of distance to the mean:

$$p(\hat v) = a \exp(\lambda(\hat \mu \cdot \hat v))$$

where $a$ is a normalization factor, $\lambda$ is a number controlling the width of the lobe (roughly corresponds to $1/2\sigma^2$ in the usual Gaussian), $\hat \mu$ is the unit direction vector for the mean of the lobe, and $\hat v$ is the unit direction vector at which we're evaluating the lobe.

(You'll also sometimes see it defined with $\lambda(\hat \mu \cdot \hat v - 1)$ in the exponent, but the $-1$ term can be pulled out and rolled into the normalization factor.)

Having a gaussian lobe usually means that your ndf is one or a weighted sum of gaussian distributions. Instead of your ndf, you can define your brdf in the same way.

A gaussian distribution is a function of the form $$\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

It has a hat shape common to many ndfs; this is good because it mimicks how normals align in most real world materials
{"ft_lang_label":"__label__en","ft_lang_prob":0.9292542,"math_prob":0.99665415,"size":1951,"snap":"2022-05-2022-21","text_gpt3_token_len":473,"char_repetition_ratio":0.10374936,"word_repetition_ratio":0.2060606,"special_character_ratio":0.23782675,"punctuation_ratio":0.08051948,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99907345,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-25T13:26:25Z\",\"WARC-Record-ID\":\"<urn:uuid:23701e87-abd0-4b1b-8007-89d90ff7ce5f>\",\"Content-Length\":\"234869\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6ed57ef3-3c3d-4f78-85d7-09c880c0be33>\",\"WARC-Concurrent-To\":\"<urn:uuid:0fe9500d-fd02-4287-b3b4-fc6fe81db797>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://computergraphics.stackexchange.com/questions/6320/what-is-a-gaussian-lobe-concerning-brdf-and-ndf/6322\",\"WARC-Payload-Digest\":\"sha1:X5TE7CAEYW5LEYAINJDLSSXVONFMDT7P\",\"WARC-Block-Digest\":\"sha1:QVFMCHELQQHQPKS5HH2L5PZF7KD5QE7F\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662587158.57_warc_CC-MAIN-20220525120449-20220525150449-00070.warc.gz\"}"}