Let $p$ be an odd prime, $k,\l$ be positive integers, $q=p^k, Q=p^{\l}$. In this work we study planar functions of the form $f_{\c}(X)=c_0X^{qQ+q}+c_1X^{qQ+1}+c_2X^{Q+q}+c_3X^{Q+1}$ over $\F_{q^2}$ for any $\c=(c_0,c_1,c_2,c_3) \in \F_{q^2}^4$. It turns out that if $f_{\c}(X)$ is planar, then $f_{\c}(X)$ is linear equivalent to one of the functions below \begin{enumerate} \item $X^{Q+1}$; \item $X^{Q+q}$; \item $P_2(x,y,)=(x^Q y, x^{Q+1}+\ep y^{Q+1})$ for some $\ep \in \F_q^*$; \item $P_3(x,y) = (x^{Q+1}-x^Q y, xy^Q+\ep y^{Q+1})$ for some $\ep \in \F_q^* \setminus \{-1\}$; \item $X^{Q+q}+\ep X^{Q+1}$ for some $\ep \in \F_{q^2}^* \setminus \mu_{q+1}$. \end{enumerate} This work is analogous to the classification of APN functions from this family $f_{\c}(X)$ for $p=2$ obtained recently by G\"{o}lo\u{g}lu. It was well-known that properties of $f_{\c}(X)$ are closely related to that of the rational function $g(X)=\frac{c_3^qX^{Q+1}+c_2^qX^Q+c_1^qX+c_0}{c_0X^{Q+1}+c_1X^Q+c_2X+c_3}$. Recently Ding and Zieve used a powerful geometric method to study permutation properties of $f_{c}(X)$ for $p=2$. The main technique of this work is to adopt their method and give a detailed study of the geometry properties of $g(X)$ for odd $p$ from which the linear equivalence follows directly.