In the present work, the distance between each marker and the Center of the image (C) is calculated using Pythagoras’ Theorem [32] (Eq 1). Since every marker has its X-Y coordinates (the centre point is (Xc, Yc), the distance of each marker concerning Centre (C) of the facial image can compute by using Pythagoras’ Theorem. The unknown ‘i’ and ‘j’ in the Eq 1 represent two points concerned to find the correspondent distance. So, the X-Y coordinates of three facial features (markers) are used to formulate one triangle (shown in Table 2). From Pythagoras’ Theorem,

Therefore, the formula for distance of each marker’s is calculated by Eq (2):

After the formulation of five triangles using eight virtual markers, an inscribed circle (Fig 5) is created inside the triangle to compute three simple statistical features, namely, Inscribed circle area of a triangle (ICAT), Inscribed Circle Circumference (ICC), and Area of a triangle (AoT). Fig 5 shows the inscribed circle of a triangle. Here, A, B, and C refer to the virtual markers. The intersection of three edges creates a center point of the circle (l). The distance between l to M_A or M_B or M_C refers to the radius of the circle (r). Here, A, B, C refers three markers used to formulate the triangle. These features are highly significant in identifying the more acceptable changes in triangle property during emotional facial expressions than other statistical features [31, 32]. Finally, these features are used for classifying different facial expressions using machine learning algorithms. The mathematical equations for deriving these features are.

The area of a triangle can be used as a simple statistical feature to detect emotions. In this work, Heron’s formula is used to calculate the area of a triangle [25], and the formula for the area of triangle computation is given in Eq (3). Here, the unknown parameters such as D1, D2, and D3 have considered three distances from the triangle.

where S is the mean value of three edge distances and defined by,

This is used to measure the circumference of the Inscribed circle [34].

where the radius (r) of the triangle is determined by,

The perimeter of the triangle can be computed as,

The area of the Inscribed circle is computed as,

where r is the radius of the inner circle.

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